Zeno Toffano
email: zeno.toffano@centralesupelec.fr

François Dubois
email: francois.dubois@math.u-psud.fr

Interpolating Binary and Multivalued Logical Quantum Gates †

Keywords: quantum gates, linear algebra, polynomial interpolation, Boolean functions, multivalued logic, quantum angular momentum

A method for synthesizing quantum gates is presented based on interpolation methods applied to operators in Hilbert space. Starting from the diagonal forms of specific generating seed operators with non-degenerate eigenvalue spectrum one obtains for arity-one a complete family of logical operators corresponding to all the one-argument logical connectives. Scaling-up to n-arity gates is obtained by using the Kronecker product and unitary transformations. The quantum version of the Fourier transform of Boolean functions is presented and a Reed-Muller decomposition for quantum logical gates is derived. The common control gates can be easily obtained by considering the logical correspondence between the control logic operator and the binary logic operator. A new polynomial and exponential formulation of the Toffoli gate is presented. The method has parallels to quantum gate-T optimization methods using powers of multilinear operator polynomials. The method is then applied naturally to alphabets greater than two for multi-valued logical gates used for quantum Fourier transform, min-max decision circuits and multivalued adders.

Introduction

Quantum calculation methods are becoming a strategic issue for emerging technologies such as quantum computing and quantum simulation. Because of the limited (at this date) quantum computational resources available it is very important to reduce the resources required to implement a given quantum circuit. The problem of optimal quantum circuit synthesis is an important topic much of the effort focuses on decomposition methods of logical gates. Quantum reversible gates have been extensively studied using principally Boolean functions implemented on Clifford, controllednot and Toffoli gates and also non-Clifford such as the 𝑇 gate. The advantages and drawbacks for privileging certain families of gates has been thoroughly investigated. Basically all methods rely on the control-logic paradigm as was first proposed by David Deutsch in [START_REF] Deutsch | Quantum theory, the Church-Turing principle and the universal quantum computer[END_REF]. The 2-qubit controlled-not gate being its simplest element resulting in the operation |𝑥, 𝑦 >→ |𝑥, 𝑥 ⊕ 𝑦 > where the exclusive disjunction (𝑋𝑂𝑅, ⊕) acts as logical negation on qubit 𝑦 when the control qubit 𝑥 is at one and leaves it unchanged otherwise. The universality of this logic is assured by the double-controlled-not or Toffoli gate with the operation on three qubits |𝑥, 𝑦, 𝑧 >→ |𝑥, 𝑦, 𝑥𝑦 ⊕ 𝑧 >. This gate operates on the third qubit 𝑧, when the conjunction is met on the control bits 𝑥 and 𝑦 (both must be 1) being thus equivalent to a negated binary conjunction 𝑁𝐴𝑁𝐷 gate, which is known to be universal. These logical controlled-gates transform the qubits as reversible permutation operators i.e., they are not diagonal in the computational basis.

In recent years many synthesizing methods for quantum circuits actually use operators that are diagonal in the computational basis because of the simpler resulting mathematical operations. The controlled-𝒁 and the double-controlled-𝒁 gates are used to build controlled-not 𝑪 and Toffoli gates 𝑻𝑶. Solutions using diagonal 𝑻 and 𝑺 gates, as will be shown hereafter, are also popular, also the stabilizer formalism and surface codes use families of Clifford diagonal gates. So in some way the "back to diagonal" trend seems to present benefits, this is also justified when one considers questions in quantum physics where much effort is employed to define Hamiltonians and the corresponding energy spectrum (eigenvalues) and stationary states (eigenstates), which can then be implemented in quantum simulation models.

A question arises: can logical calculations be formalized someway directly in the qubit eigenspace? The answer seems to be affirmative and in a rather simple way too. A proposal has been given by making parallels between propositional logic and operator linear algebra in the framework of "Eigenlogic" [START_REF] Toffano | Eigenlogic in the spirit of George Boole[END_REF][START_REF] Dubois | A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF], the idea is that Boolean algebra operations can be represented by operators in Hilbert space, using projection operators as developed in [START_REF] Toffano | Eigenlogic in the spirit of George Boole[END_REF] and extended to angular momentum operators as shown in [START_REF] Dubois | A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] and generalized in this paper.

A theoretical justification could be inspired by Pierre Cartier in [START_REF] Cartier | A mad day's work: Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry[END_REF], relating the link between the algebra of logical propositions and the set of all valuations on it, he writes: "...in the theory of models in logic a model of a set of propositions has the effect of validating certain propositions. With each logical proposition one can associate by duality the set of all its true valuations represented by the number 1. This correspondence makes it possible to interpret the algebra of propositions as a class of subsets, conjunction and disjunction becoming respectively the intersection and union of sets. This corresponds to the Stone duality proved by the Stone representation theorem and is one of the spectacular successes of twentieth century mathematics.... The development of quantum theory led to the concept of a quantum state, which can be understood as a new embodiment of the concept of a valuation". The idea is not new, and stems from John Von Neumann's proposal of "projections as propositions" in [START_REF] Neumann | Mathematische Grundlagen der Quantenmechanik[END_REF] which was subsequently formalized in quantum logic with Garret Birkhoff in [START_REF] Birkhoff | The Logic of Quantum Mechanics[END_REF]. These topics have been thoroughly discussed in [START_REF] Toffano | Eigenlogic in the spirit of George Boole[END_REF].

But because nowadays quantum logic is mostly interested in aspects concerning operations going beyond and also in contrast with the principles of classical logic it is still not considered as an operational tool for quantum computing even though many bridges have been made [START_REF] Bub | Quantum computation from a quantum logical perspective[END_REF][START_REF] Ying | A theory of computation based on quantum logic (I)[END_REF]. The aim of Eigenlogic is on the other side to fully exploit the logical structure offered by the operational system in the eigenspace with of course the possibility to look outside at other basis representations this leads for example to fuzzy logic applications [START_REF] Dubois | A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] as will be explicated hereafter.

While most of the research is currently devoted to quantum circuit and algorithm developments based on the use of quantum operators working in binary (Boolean) systems through the manipulation of qubits, there are many possibilities for the exploitation of observables with more than two non-degenerate eigenvalues and the theory of multi-valued-logic is very naturally applicable to the design and analysis of these systems.

Eigenlogic Interpretation

Historical Background

George Boole gave a mathematical symbolism through the two numbers {0,1} representing resp. the "false" or "true" character of a proposition [START_REF] Boole | The Mathematical Analysis of Logic: Being an Essay to a Calculus of Deductive Reasoning[END_REF] (see [START_REF] Toffano | Eigenlogic in the spirit of George Boole[END_REF] for an historical discussion). An idempotent symbol 𝑥 verifies the equation: 𝑥 2 = 𝑥 which admits only two possible solutions: 0 and 1, this equation was considered by Boole the "fundamental law of thought". As will be emphasized here the algebra of idempotent symbols can also be interpreted as a set of commuting projection operators in linear algebra.

John von Neumann in [START_REF] Neumann | Mathematische Grundlagen der Quantenmechanik[END_REF] was the first to associate projection operators with logical propositions, which subsequently was formalized in an independent discipline: quantum logic [START_REF] Birkhoff | The Logic of Quantum Mechanics[END_REF]. He also introduced the formalism of the density matrix in quantum mechanics where a pure quantum state |𝜓⟩ can also be represented by a projection operator: 𝝆 = |𝜓⟩⟨𝜓|, which is a ray (a rank-1 idempotent projection operator spanning a one-dimensional subspace). All these concepts lay at the foundations of quantum theory.

Binary Eigenlogic

A linear algebraic method is presented here, for binary, multi-valued and fuzzy logic using observables in Hilbert space. All logical connectives have their corresponding observable where the truth values correspond to eigenvalues. In this way propositional logic can be formalized by using tensor combinations of elementary quantum observables. The outcome of a "measurement" of a logical observable will give the truth value of the associated logical proposition, and becomes thus interpretable when applied to vectors of its eigenspace, leading to an original insight into the quantum measurement postulate.

A projection operator in Hilbert space is associated to a logical proposition, the operator being Hermitian it has the properties of a quantum observable and is considered here a logical observable. This view is named Eigenlogic [START_REF] Toffano | Eigenlogic in the spirit of George Boole[END_REF][START_REF] Dubois | A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] and can be summarized as follows:

 eigenvectors in Hilbert space ⟺ interpretations (atomic propositional cases)  logical observables ⟺ logical connectives  eigenvalues ⟺ truth values
The general form of a logical observable can be expressed as a development:

𝑭 = 𝑓(0)𝜫 0 + 𝑓(1)𝜫 1 = 𝑑𝑖𝑎𝑔[𝑓(0), 𝑓 (1)]
the terms in the development are the 2-dimensional rank-1 projectors 𝜫 0 and 𝜫 1 , the cofactors 𝑓(0) and 𝑓(1) are the eigenvalues corresponding to the truth values taking the values {0,1}. This allows to generate the 4 one-argument logical observables:

𝑭 𝐴 = 0𝜫 0 + 1𝜫 1 = 𝜫 , 𝑭 𝐴 ̅ = 1𝜫 0 + 0𝜫 1 𝑭 ⊥ = 0𝜫 0 + 0𝜫 1 = 𝕆 2 , 𝑭 𝑇 = 1𝜫 0 + 1𝜫 1 = 𝑰 d
these are represented by the logical projector 𝐴, its negation 𝐴 ̅ , contradiction ⊥ and tautology 𝑇.

Negation corresponds to complementation obtained by subtracting from the identity operator 𝑰 d :

𝑭 ̅ = 𝑰 d -𝑭
For the one-argument case, 𝑛 = 1, the eigenvectors |0⟩ and |1⟩ correspond to the 2-dimensional orthonormal vectors:

|0⟩ = ( 1 0 ) , |1⟩ = ( 0 1 )
Vectors |0⟩ and |1⟩, form the canonical basis, and correspond in quantum mechanics to qubits corresponding respectively to the north and south polar points on the Bloch sphere (see Figure 1).

The two-argument logical observables can be developed in a similar way:

𝑭 2 = 𝑑𝑖𝑎𝑔[𝑓(0,0), 𝑓(0,1), 𝑓(1,0), 𝑓(1,1)]
There are 2 2 𝑛 logical connectives for a 𝑛-argument (arity) system. For 𝑛 = 1 this gives the four connectives described above. For 𝑛 = 2 one has 16 binary logical connectives: conjunction (∧, 𝐴𝑁𝐷); disjunction (∨, 𝑂𝑅); exclusive disjunction (⨁, 𝑋𝑂𝑅); logical projectors (A, B); material implication ⇒; Sheffer stroke (↑, 𝑁𝐴𝑁𝐷); tautology (T). The remaining ones can be derived by the classical theorems of logic. All logical connectives are uniquely characterized by their truth table i.e., by their logical semantics. The complete family of the 16 two-argument connectives are shown on Table 1. For two arguments eigenvectors are named |𝑎 𝑏⟩ where the arguments 𝑎 and 𝑏 take the values {0,1} and represent one of the four possible cases (logical interpretations) for the input of the logical connective, they are calculated using the Kronecker product of the qubit state vectors |0⟩ and |1⟩, explicitly:

|00⟩ = |0⟩⨂|0⟩ = (1,0,0,0) 𝑡 , |01⟩ = |0⟩⨂|1⟩ = (0,1,0,0) 𝑡 |10⟩ = |1⟩⨂|0⟩ = (0,0,1,0) 𝑡 , |11⟩ = |0⟩⨂|0⟩ = (0,0,0,1) 𝑡
Using the following linear bijection between the projection observable 𝑭 and a reversible observable 𝑮:

𝑮 = 𝑰 d -2𝑭
one obtains a logical binary system using now the numbers {+1, -1}. Observables 𝑭 and 𝑮 commute, so they have the same system of eigenvectors. The eigenvalues that correspond to the truth values, respectively "false" and "true", are respectively +1 and -1.

In this case the generating observable, the seed operator, is the Pauli matrix 𝜎 𝑧 also named the 𝒁 gate:

𝜎 𝑧 = 𝒁 = 𝐈 d -2𝜫 1 = ( +1 0 0 -1 )
For two arguments, in the system {+1, -1}, the logical dictators 𝑼 and 𝑽 (the equivalent of the logical projectors 𝑨 and 𝑩 for the system {0,1}) are function of 𝒁 by: 𝑼 = 𝒁⨂𝐈 d = 𝑑𝑖𝑎𝑔(+1, +1, -1, -1) , 𝑽 = 𝐈 d ⨂𝒁 = 𝑑𝑖𝑎𝑔(+1, -1, +1, -1) Negation in the system {+1, -1} is obtained by multiplying by -1:

𝑮 ̅ = -𝑮

This formulation leads to the same results as for the Fourier transform model of Boolean functions developed in [START_REF] O'donnell | Analysis of Boolean Functions[END_REF] and extended to the quantum case in [START_REF] Montanaro | Quantum Boolean Functions[END_REF].

Fuzzy Eigenlogic

What happens when the quantum state is not one of the eigenvectors of the logical system? In quantum mechanics one can always express a quantum state-vector |𝜓⟩ as a combination on a complete orthonormal basis, in particular on the canonical eigenbasis of the logical observable family, this gives:

|𝜓⟩ = 𝑐 00 |00⟩ + 𝑐 01 |01⟩+𝑐 10 |10⟩ + 𝑐 11 |11⟩
When only one of the coefficients is not zero, then one obtains a fixed interpretation for the proposition corresponding to one of the eigenstates and goes back to the preceding discussion.

Fuzzy logic deals with truth values that can take values between 0 and 1, so the truth of a proposition can lie between "completely true" and "completely false". When more than one coefficient in the development of |𝜓⟩ is non-zero one can consider a fuzzy interpretation.

For a projective observable 𝑭 measured in the context of the quantum state |𝜓⟩ the mean value (Born rule) gives directly a probability measure by:

𝑝 |𝜓⟩ = ⟨𝜓|𝑭|𝜓⟩ = 𝑇𝑟(𝝆 • 𝑭) with 𝝆 = |𝜓⟩⟨𝜓| the density matrix
The obtained mean value of the logical observable 𝑭 is thus a fuzzy measure of the truth of a logical proposition in the form of a fuzzy membership function 𝜇.

For one-argument an arbitrary 2-dimensional quantum state can be written as:

|𝜙⟩ = sin𝛼|0⟩ + 𝑒 𝑖𝛽 cos𝛼 |1⟩
where the angles 𝛼 and 𝛽 are real numbers. The membership function is the quantum mean value of the logical projector observable 𝑨 = 𝚷, given by:

𝜇(𝑎) = ⟨𝜙|𝚷|𝜙⟩ = cos 2 𝛼
Using the angular transformation 𝛼 = 𝜃/2 and 𝛽 = 𝜑/2 one can represent the angles 𝜃 and 𝜑 on the Bloch sphere (see Figure 1).

A quantum compound state can be built by taking the tensor product of two elementary states: One can calculate the fuzzy membership function of the corresponding "logical projector" for the two-argument case:

|𝜓 ≥ | 𝜑 𝑝 > ⊗ | 𝜑 𝑞 >,
𝜇(𝑎) =< 𝜓|𝜫 ⊗ 𝑰|𝜓 >= 𝑝(1 -𝑞) + 𝑝 ⋅ 𝑞 = 𝑝 , 𝜇(𝑏) =< 𝜓|𝑰 ⊗ 𝜫|𝜓 >= 𝑞
this shows that the mean values correspond to the respective probabilities. Now let's "measure" for example the conjunction and the disjunction, using the observables in table 1, this gives:

𝜇(𝑎 ∧ 𝑏) =< 𝜓 |𝜫 ⊗ 𝜫|𝜓 >= 𝑝 ⋅ 𝑞 = 𝜇(𝑎) ⋅ 𝜇 (𝑏) 𝜇(𝑎 ∨ 𝑏) = 𝑝 + 𝑞 -𝑝 ⋅ 𝑞 = 𝜇(𝑎) + 𝜇(𝑏) -𝜇(𝑎) ⋅ 𝜇(𝑏)
So in this approach fuzzy logic arises naturally when considering vectors outside the eigensystem. The fuzzy membership function is obtained by the quantum mean value (Born rule) of the logical projection observable and turns out to be a probability measure. Fuzziness arises because of the quantum superposition of interpretations, the truth of a proposition being in this case a probabilistic value ranging from completely false to completely true.

Multivalued Eigenlogic

The recently observed revival of interest in applying multi-valued logic (MVL) to the description of quantum phenomena is closely related to fuzzy logic. MVL is of interest to engineers involved in various aspects of information technology and has a long history, for example it is at the basis of the programming language HDL (Hardware Description Language).

In MVL the total number of logical connectives for a system of 𝑚 values and 𝑛 arguments is the combinatorial number 𝑚 𝑚 𝑛 , giving for example for a 3-valued 2-argument system 3 3 2 = 19,683 different logical connectives. This shows that using the totality of the different logical connectives in a single mathematical framework becomes intractable, but some special connectives play an important role.

MVL is naturally associated to the physical quantity: quantum angular momentum. The obseravble of the 𝑧 component of the angular momentum observable for ℓ = 1 is:

𝑳 𝑧 = ℏ 𝑑𝑖𝑎𝑔(+1,0, -1)
one can associate the eigenvalues to the logical truth values: "false" F ≡ +1 , "neutral" N ≡ 0 , "true" T ≡ -1

In this case, as will be shown using the method presented hereafter, the logical observables can be expressed as a unique development by spectral decompositions over rank-1 projection operators:

The Interpolation Method for Quantum Operators

Lagrange and Cayley-Hamilton Interpolation

The method presented here is inspired from classical Lagrange interpolation where the "variable" is represented by a seed operator acting in Hilbert space, possessing 𝑚 distinct eigenvalues, i.e., it is non-degenerate. The values are not fixed meaning that one can work with different alphabets, in the binary case one uses classically the Booleans {0,1}, but also {+1, -1} can be considered. What this method will show is that for whatever finite system of values, unique logical operators can be defined. In the multivalued case, popular choices are, for example {0,1,2, … , 𝑚 -1} formalized in Post logic [START_REF] Post | Introduction to a General theory of Elementary Propositions[END_REF] and rational fractional values in the unit interval [0,1] giving the system {0, The possibility of using complex values can also be considered and this has applications for specific problems such as in 𝑻-gate synthesis and quantum Fourier transform in this last case one considers the roots of unity giving the 𝑚-eigenvalue spectrum: {𝑒 𝑖2𝜋0 , 𝑒 𝑖2𝜋 1 𝑚 , 𝑒 𝑖2𝜋 2 𝑚 , … , 𝑒 𝑖2𝜋 𝑚-1 𝑚 }. In each situation one starts by defining the seed operator 𝜦 with 𝑚 non-degenerate eigenvalues λi. The density matrix of an eigenstate |𝜆 𝑖 > is 𝜫 𝜆 𝑖 = |𝜆 𝑖 > < 𝜆 𝑖 |, and corresponds to a rank-1 projection operator, a ray, for this state. It is obtained by the following expression:

𝜫 𝜆 𝑖 (𝜦) = ∏ 𝜦 -𝜆 𝑗 𝑰 𝑚 𝜆 𝑖 -𝜆 𝑗 𝑚 𝑗=1,𝑗≠𝑖
The development is unique according to the Cayley-Hamilton theorem, this expression can be put in polynomial form consisting in a linear combination of powers up to 𝑚 -1 of the seed operator 𝜦, it is a rank-1 projection operator over the 𝑚 dimensional Hilbert space and can be expressed as a 𝑚 × 𝑚 square matrix. All the rank-1 projection operators obtained in the eigensystem commute and span the entire space by the closure relation:

∑ 𝜫 𝜆 𝑖 𝑚 𝑖=1 = ∑|𝜆 𝑖 > < 𝜆 𝑖 | = 𝑰 𝑚 𝑚 𝑖=1
as outlined in [START_REF] Toffano | Eigenlogic in the spirit of George Boole[END_REF][START_REF] Dubois | A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] in Eigenlogic eigenvalues correspond to logical truth values (values {0,1} for a Boolean system) and one can make the correspondence between propositional logical connectives and operators in Hilbert space. The truth table of a logical connective corresponds to different semantic interpretations, each interpretation is a fixed attribution of truth values to the elementary propositions (the "inputs") composing the connective. In Eigenlogic each interpretation corresponds to one of the eigenvectors and the associated eigenvalue to the corresponding truth-value for the considered logical connective.

It is well known that logical connectives can be expressed through arithmetic expressions, these are closely related to polynomial expressions over rings, but with expressions using ordinary arithmetic addition and subtraction instead of their modular counterpart as used in Boolean algebra. These topics were thoroughly discussed in [START_REF] Toffano | Eigenlogic in the spirit of George Boole[END_REF] for the Boolean values {0,1}. Arithmetic developments of logical connectives are often used, for example, in the description of switching functions for decision logic design. A good review is given in [START_REF] Yanushkevich | Introduction to Logic Design[END_REF].

In logic, functions and arguments take the same values. For Boolean functions the possible values are the two numbers 0 and 1 corresponding respectively to the false and true character of a logical proposition. So considering an arithmetic expression for an arity-one logical connective ℓ(𝑎) and choosing the 𝑚 distinct logical values 𝑎 𝑖 , the value taken by the logical function at one of these points 𝑎 𝑝 , is ℓ(𝑎 𝑝 ) ∈ {𝑎 1 , 𝑎 2 , . . . , 𝑎 𝑚 }, also one of these values. The corresponding unique logical operator is given by the interpolation development:

𝑭 ℓ = ∑ ℓ(𝑎 𝑖 ) 𝜫 𝑎 𝑖 𝑚 𝑖=1
this operator decomposition is proved by Sylvester's theorem and represents the spectral decomposition of the operator. The projection operators are obtained by the interpolation formula with 𝜆 𝑖 = 𝑎 𝑖 .

One must then consider the operators corresponding to the elementary propositions of the logical connectives. This is a straightforward procedure in Eigenlogic. For arity-one logical connectives, these are function of one single elementary proposition which corresponds, in the method proposed here, to the seed operator 𝜦. These concepts will become clearer with some examples.

Scaling to higher arity for more logical arguments

The scaling is obtained by following the same procedure as for classical interpolation methods for multivariate systems using tensor products (see e.g., [START_REF] Kikuchi | Finite Element Methods in Mechanics[END_REF]). Here the chosen convention for the indexes is the one given by David Mermin in [START_REF] Mermin | Quantum Computer Science: An Introduction[END_REF] where index 0 indicates the lowest digit, index 1 the next and so on... For arity-2 one considers two operators 𝑷 1 and 𝑸 0 corresponding to two elementary propositional variables. One can write by the means of the Kronecker product ⊗: 𝑷 1 = 𝜦 ⊗ 𝑰 , 𝑸 0 = 𝑰 ⊗ 𝜦 for arity-3 using three operators one has:

𝑷 2 = 𝜦 ⊗ 𝑰 ⊗ 𝑰 , 𝑸 1 = 𝑰 ⊗ 𝜦 ⊗ 𝑰 , 𝑹 0 = 𝑰 𝑰 ⊗ 𝜦
for higher arity-𝑛 the procedure can be automatically iterated.

The logical operators can then be calculated, these are multi-linear combinations of the elementary operators 𝑷 𝑖 , 𝑸 𝑗 . For example the simplest operators are products. Due to logical completeness (demonstrated by Emil Post in [START_REF] Post | Introduction to a General theory of Elementary Propositions[END_REF]) there will be for an 𝑚-valued 𝑛-arity system exactly 𝑚 𝑚 𝑛 logical operators forming a complete family of commuting logical operators. Logical completeness has also another important consequence: there are always universal connectives from which all the others can be derived, it has been shown, also in [START_REF] Post | Introduction to a General theory of Elementary Propositions[END_REF], that for an m-valued arity-2 system one needs at least two universal connectives, in the case of Post logic these turn out to be the general negation (cyclic permutation of all values 𝑎 𝑖 → 𝑎 𝑖+1 ) and the 𝑀𝑎𝑥 connective (takes the highest of the two input values) these reduce to Boolean connectives for binary values where the 𝑀𝑎𝑥 connective becomes the disjunction (𝑂𝑅, ∨). The logical operators obtained here for an 𝑚-valued 𝑛-arity system are represented by 𝑚 𝑛 × 𝑚 𝑛 square matrices.

Building the Controlled-Not and Toffoli Binary Gates

One Qubit Gates

In the computational basis the 𝒁 and 𝑿 and Hadamard 𝑯 gate's matrix forms are:

𝒁 = ( +1 0 0 -1 ) , 𝑿 = ( 0 1 1 0 ) , 𝑯 = 1 √2 ( +1 1 1 -1 )
One has to consider the useful opeerator transformation 𝑿 = 𝑯 ⋅ 𝒁 ⋅ 𝑯 and also 𝒁 = 𝑯 ⋅ 𝑿 ⋅ 𝑯 (because 𝑯 2 = 𝑰):

The seed operator for the system {+1, -1} in the computational basis is:

𝜦 { + 1, - 1} = 𝒁 = 𝑑𝑖𝑎𝑔 𝑧 ( + 1, - 1)
The one-qubit state vectors and projection operators are:

|0 > = ( 1 0 ) , |1 > = ( 1 0 ) 𝜫 +1 (𝒁) = 1 2 (𝑰 + 𝒁) = |0 > < 0| , 𝜫 -1 (𝒁) = 1 2 (𝑰 -𝒁) = |1 > < 1| = 𝜫
The same method can also be used when operators are not diagonal in the computational basis, for example one could have chosen the 𝑿 gate as the seed operator by changing 𝒁 into 𝑿.

The seed operator and the two projection operators permit to write the four arity-one logical operators: 𝒁 (Dictator), - 𝒁 (Negation), + 𝑰 (Contradiction) and - 𝑰 (Tautology) (see [START_REF] Toffano | Eigenlogic in the spirit of George Boole[END_REF][START_REF] Dubois | A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] for a detailed discussion), the eigenvalues {+1, -1} correspond respectively to False and True.

Before continuing for higher arity it is interesting to analyze the logical operator system corresponding to the Boolean values {0,1}, it is straightforward to see that in this case the seed operator, 𝚲 {0, 1} , is the projection operator 𝜫 1 = 𝜫 given above, the other projection operator Π0 being the complement:

𝚲 {0, 1} = 𝑑𝑖𝑎𝑔 𝑧 (0, 1) = 𝜫 1 = 𝜫 = |1 > < 1|
, 𝜫 0 = 𝑰 -𝜫 = |0 > < 0|

There are interesting tranformations for the operators when going from the system {+1, -1} to the system {0,1}. Due to the properties of the operator Π (idempotent projection operator 𝜫 2 = 𝜫) and 𝒁 (self-inverse operator 𝒁 2 = 𝑰) one can use the Householder transform:

𝒁 = 𝑰 -2 𝜫 = ( - 1) 𝜫 = 𝑒 𝑖𝜋𝜫 = 𝑒 𝑖 𝜋 2 𝑒 -𝑖 𝜋 2 𝒁 , 𝒁 |𝑥 > = ( - 1) 𝑥 |𝑥 >
This transform applies also for composite expressions and permits to transform operators with eigenvalues {0,1} into operators with eigenvalues {+1, -1} and will be used hereafter to build the controlled-not 𝑪 and Toffoli gates 𝑻𝑶.

For arity-2, one considers the two elementary operators acting on qubit-0 and qubit-1:

𝒁 𝟎 = 𝑰 ⊗ 𝒁 , 𝒁 𝟏 = 𝒁 ⊗ 𝑰

Building Cz and CNOT gates

The controlled-z gate, named here 𝑪 𝑍 , is diagonal in the computational basis. The interpolation method described above and used in [START_REF] Dubois | A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] gives directly the known [START_REF] Mermin | Quantum Computer Science: An Introduction[END_REF] polynomial expression:

𝑪 𝒁 = 𝑑𝑖𝑎𝑔 𝑧 ( + 1, + 1, + 1, - 1) = 1 2 ( 𝑰 + 𝒁 𝟏 + 𝒁 𝟎 -𝒁 𝟏 ⋅ 𝒁 𝟎 )
One can consider some interesting interpretations using Eigenlogic [START_REF] Dubois | A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF]: the truth table is given by the structure of the eigenvalues of the 𝑪 𝒁 operator and corresponds to the logical connective conjunction (AND, ∧), where the values for {False, True} correspond here to the alphabet {+ 1, - 1}. Also the logical operator for conjunction 𝑭 𝐴𝑁𝐷 in the Boolean alphabet, {𝑥, 𝑦} ∈ {0, 1} , as demonstrated in [START_REF] Toffano | Eigenlogic in the spirit of George Boole[END_REF], has a very simple form: The controlled-not operator, named here 𝑪, can be expressed straightforwardly, see e.g., [START_REF] Mermin | Quantum Computer Science: An Introduction[END_REF], in its polynomial form on 𝒁 1 and 𝑿 0 :

𝑭 𝐴𝑁𝐷 = 𝜫 ⊗ 𝜫 = 𝑑𝑖𝑎𝑔 𝑧 (0, 0, 0,
𝑪 = 1 2 (𝑰 + 𝒁 1 + 𝑿 0 -𝒁 𝟏 ⋅ 𝑿 0 )
this expression can be derived from 𝑪 𝒁 using the transformation 𝑿 0 = 𝑯 0 ⋅ 𝒁 0 ⋅ 𝑯 0 , b where the operator 𝑯 0 = 𝑰 ⊗ 𝑯 with 𝑯 is the Hadamard gate defined above.

In the preceding section it has been emphasized that one can consider different basis to define the system and the respective seed operator.

Some important properties about projection operators (not necessarily commuting) have to outlined:

(i) The Kronecker product of two projection operators is also a projection operator. (ii) If projection operators are rank-1 (a single eigenvalue is 1 all the others are 0) then their Kronecker product is also a rank-1 projection operator.

So for example one can use the computational basis (𝒁 eigenbasis) for one qubit and the 𝑿 eigenbasis for the other qubit. This is used for expressing the operator form of the controlled-not 𝑪 gate as function of the projection operators. Taking the analogy with the 𝑪 𝒁 gate one can define the projection operator associated to the 𝑪 gate as:

𝜫 𝑪 = 𝜫 ⊗ 𝜫 𝑿 = ( 0 0 0 1 ) ⊗ 1 2 ( 1 -1 -1 1 ) = 1 2 ( 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 1 )
where the projection operator on the 𝑿 eigenbasis is easily derived using 𝜫 𝑿 = 𝑯 ⋅ 𝜫 ⋅ 𝑯. Then one obtains straightforwardly the well-known matrix:

𝑪 = 𝑰 -2𝜫 𝑪 = ( - 1) 𝜫⊗𝜫 𝑿 = ( 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 )

Building the TOFFOLI Gate

The same method can be applied to build the doubly-controlled not-gate (the Toffoli gate 𝑻𝑶). One starts again with the analogy with conjunction and notices that the gate uses 3 qubits in a 8dimensional space, the compound state in the computational basis is |xyz >. In the same way as before one can define here a doubly-controlled-𝒁 gate 𝑪 𝑪𝒁 :

(35) 𝑪 𝑪𝒁 = 𝑰 -2(𝜫 ⊗ 𝜫 ⊗ 𝜫) = ( - 1) 𝜫⊗𝜫⊗𝜫 = 𝑑𝑖𝑎𝑔 𝑧 (1, 1, 1, 1, 1, 1, 1, - 1) 𝑪 𝑪𝒁 |𝑥𝑦𝑧 > = ( - 1) 𝑥𝑦𝑧 |𝑥𝑦𝑧 >
Then the 𝑻𝑶 gate can be found by the same method as for the 𝑪 gate. The polynomial expression is easily calculated giving:

𝑻𝑶 = 𝑯 0 ⋅ 𝑪 𝑪𝒁 ⋅ 𝑯 0 = 𝑰 -2(𝜫 ⊗ 𝜫 ⊗ 𝜫 𝑿 ) = 1 2 (𝑰 + 𝒁 2 + 𝑪 -𝒁 2 ⋅ 𝑪)
The 𝑪 gate can be expanded giving the alternative expression as function of single qubit gates:

𝑻𝑶 = 1 4 (3 𝑰 + 𝒁 2 + 𝒁 1 + 𝑿 0 -𝒁 2 ⋅ 𝒁 1 -𝒁 2 ⋅ 𝑿 0 -𝒁 1 ⋅ 𝑿 0 + 𝒁 2 ⋅ 𝒁 1 ⋅ 𝑿 0 )
In quantum circuits it is practically difficult to realize the sum of operators and one prefers, if it is possible, to use a product form representing the same operator. Using the self-inverse symmetry of the above operators it is a standard procedure to make this transformation [START_REF] Mermin | Quantum Computer Science: An Introduction[END_REF] using the Householder transform. The 𝑪 𝒁 gate polynomial expression can be transformed in the following way:

𝑪 𝒁 = ( - 1) 𝜫 𝑪 𝒁 = 𝑒 𝑖 𝜋𝜫 𝑪 𝒁 = 𝑒 𝑖 𝜋 2 𝑒 -𝑖 𝜋 4 ( 𝑰 + 𝒁 𝟏 + 𝒁 𝟎 - 𝒁 𝟏 ⋅ 𝒁 𝟎 ) = 𝑒 𝑖 𝜋 4 𝑒 -𝑖 𝜋 4 𝒁 𝟏 𝑒 -𝑖 𝜋 4 𝒁 𝟎 𝑒 +𝑖 𝜋 4 𝒁 𝟏 ⋅ 𝒁 𝟎
The factorization of the exponential operators is allowed because all the argument operators in the exponential commute, the order of the multiplication can thus be interchanged.

The same method can be used to obtain an expression for the controlled-not 𝑪 gate, one just replaces 𝒁 0 by 𝑿 0 .

In the same way this leads to a new factorized expression of the Toffoli gate 𝑻𝑶:

𝑻𝑶 = 𝑒 𝑖 𝜋 8 𝑒 -𝑖 𝜋 8 𝒁 𝟐 𝑒 -𝑖 𝜋 8 𝒁 𝟏 𝑒 -𝑖 𝜋 8 𝑿 0 𝑒 +𝑖 𝜋 8 𝒁 𝟐 ⋅ 𝑿 0 𝑒 +𝑖 𝜋 8 𝒁 𝟐 ⋅ 𝒁 𝟏 𝑒 +𝑖 𝜋 8 𝒁 1 ⋅𝑿 0 𝑒 -𝑖 𝜋 8 𝒁 2 ⋅𝒁 1 ⋅𝑿 0
this formulation shows also that it is easy to scale up the gates for example with a Toffoli-4 gate using three control bits on a 4 qubit state |𝑥𝑦𝑧𝑤 >.

Correspondence with Recent T-Gate Based Methods

There has been much interest recently for developing general methods for synthesizing quantum gates based on polynomial methods [START_REF] Muller | Application of Boolean algebra to switching circuit design and to error detection[END_REF][START_REF] Amy | T-count optimization and Reed-Muller codes[END_REF][START_REF] Selinger | Quantum circuits of T-depth one[END_REF]. The decomposition of arbitrary gates into Clifford and 𝑇-set gates is an important problem. It is often desirable to find decompositions that are optimal with respect to a given cost function. The exact cost function used is application dependent; some possibilities are: the total number of gates; the total number of T gates; the circuit depth and/or the number of ancillas used.

The single-qubit non-Clifford gate 𝑻 and Clifford gate 𝑺 are derived from the 𝒁-gate and are expressed in the computational basis in their matrix form:

𝑻 = 𝒁 1 4 = ( 1 0 0 𝑒 𝑖 𝜋 4 ) , 𝑺 = 𝒁 1 2 = ( 1 0 0 𝑒 𝑖 𝜋 4 ) = ( 1 0 0 𝑖 )
as stated above these operators are diagonal in the computational basis.

Considering the preceding discussion the seed operator 𝜦 for a 𝑇-set is the 𝑻-gate itself with eigenvalues {+1, 𝜔}, naming 𝜔 = 𝑒 𝑖 𝜋 4 . The action of the 𝑻-gate on a quibit in the computational basis is: 𝑻 |𝑥 ≥ 𝜔 𝑥 |𝑥 >.

One can also define the conjugate transpose gate

𝑻 † |𝑥 > = (𝜔 † ) 𝑥 |𝑥 > = 𝑒 -𝑖 𝜋 4 𝑥 |𝑥 > = 𝜔 -𝑥 |𝑥 > .
Given the two following arithmetic expressions of exclusive disjunction, (𝑋𝑂𝑅, ⊕), for 2 and 3 Boolean arguments [START_REF] Toffano | Eigenlogic in the spirit of George Boole[END_REF]:

𝑥 ⊕ 𝑦 ⊕ 𝑧 = 𝑥 + 𝑦 + 𝑧 -2𝑥𝑦 -2𝑥𝑧 -2𝑦𝑧 + 4𝑥𝑦𝑧 , 𝑥 ⊕ 𝑦 = 𝑥 + 𝑦 -2𝑥𝑦
where the second member is an inclusion-exclusion-like form and combining the expressions gives:

4𝑥𝑦𝑧 = 𝑥 + 𝑦 + 𝑧 -𝑥 ⊕ 𝑦 -𝑥 ⊕ 𝑧 -𝑦 ⊕ 𝑧 + 𝑥 ⊕ 𝑦 ⊕ 𝑧
this last expression gives a method for building more complex gates using only 𝑻 and 𝑻 † gates as shown by Peter Selinger in [START_REF] Selinger | Quantum circuits of T-depth one[END_REF]. Starting again with the double controlled-Z gate 𝑪 𝑪𝒁 one uses 𝑻 = 𝒁 1 4 = 𝑒 𝑖 𝜋 8 𝑒 -𝑖 𝜋 8 𝒁 .

Defining the 3-qubit operators:

𝑻 0 = 𝑰 ⊗ 𝑰 ⊗ 𝑻 , 𝑻 1 = 𝑰 ⊗ 𝑻 ⊗ 𝑰 , 𝑻 2 = 𝑻 ⊗ 𝑰 ⊗ 𝑰
Using Reed-Muller decompositions [START_REF] Yanushkevich | Introduction to Logic Design[END_REF][START_REF] Muller | Application of Boolean algebra to switching circuit design and to error detection[END_REF][START_REF] Amy | T-count optimization and Reed-Muller codes[END_REF] one can express the 𝑪 𝑪𝒁 operator using the action on the 3-qubit state |𝑥𝑦𝑧 > and the eigenvalue relation (-1) 𝑥 = 𝜔 4𝑥 , leading to [START_REF] Selinger | Quantum circuits of T-depth one[END_REF]: [3] ) † ⋅ (𝑻 𝑥⊕𝑧 [3] ) † ⋅ (𝑻 𝑦⊕𝑧 [3] ) † ⋅ (𝑻 𝑥⊕𝑦⊕𝑧 [START_REF] Dubois | A Quantum View for Multiple-Valued and Fuzzy Systems[END_REF] ) the operators corresponding to exclusive disjunction ⊕ can be easily obtained using the Eigenlogic interpretation: they are diagonal operators where the diagonal elements are the truth values of the corresponding logical connective using the 𝑻 operator alphabet {+1, 𝜔}. For example explicitly:

𝑪 𝑪𝒁 = 𝑻 0 ⋅ 𝑻 1 ⋅ 𝑻 2 ⋅ (𝑻 𝑥⊕𝑦
𝑻 𝑥⊕𝑦 [3] = 𝑻 𝑥⊕𝑦 [2] ⊗ 𝑰 = 𝑑𝑖𝑎𝑔 𝑧 (1, 𝜔, 𝜔, 1) ⊗ 𝑰 = 𝑑𝑖𝑎𝑔 𝑧 (1, 1, 𝜔, 𝜔, 𝜔, 𝜔,

It can be shown, again because 𝑻 = 𝑒 𝑖 𝜋 8 𝑒 -𝑖 𝜋 8 𝒁 , that the Toffoli gate 𝑻𝑶 = 𝑯 0 ⋅ 𝑪 𝑪𝒁 ⋅ 𝑯 0 is equivalent to the expression function of 𝒁 and 𝑿 given above. An alternative polynomial expression can be found directly by the interpolation method. The idea is that because 𝑻 and 𝒁 commute and are not degenerate they have the same eigenvectors and thus one can use the same projection operators 𝜫 and its complement 𝑰 -𝜫. The expression of the double controlled-𝒁 gate 𝑪 𝑪𝒁 as a function of 𝜫 has been calculated above, now one just has to express the projection operator 𝜫 as function of the operator 𝑻 by the interpolation equation:

𝜫 𝜔 (𝑻) = 𝜫 = (𝜔 -1) -1 ( 𝑻 -𝑰) , 𝜫 +1 (𝑻) = 𝑰 -𝜫 = - (𝜔 -1) -1 ( 𝑻 -𝜔𝑰)
so directly:

𝑪 𝑪𝒁 = 𝑰 -2(𝜫 ⊗ 𝜫 ⊗ 𝜫) = 𝑰 -2(𝜔 -1) -3 [(𝑻 -𝑰) ⊗ (𝑻 -𝑰) ⊗ (𝑻 -𝑰)]
which can also be expressed as a function of 𝑻 0 , 𝑻 1 and 𝑻 2 :

𝑪 𝑪𝒁 = 𝑰 + 2(𝜔 -1) -3 (𝑰 -𝑻 2 -𝑻 1 -𝑻 0 + 𝑻 2 ⋅ 𝑻 1 + 𝑻 2 ⋅ 𝑻 0 + 𝑻 1 ⋅ 𝑻 0 -𝑻 2 ⋅ 𝑻 1 ⋅ 𝑻 0 )
again using the transformation method it is easy to show that this expression is equivalent to the preceding ones. The Toffoli gate 𝑻𝑶 is then straightforwardly derived by replacing 𝑻 𝟎 by (𝑯 0 ⋅ 𝑻 𝟎 ⋅ 𝑯 0 ) in the expression of 𝑪 𝑪𝒁 .

The same method could be employed using 𝑺 gates, for example simply by replacing the 𝑻 operators by the respective 𝑺 ones and 𝜔 = 𝑒 𝑖 𝜋 4 by 𝜔 𝑆 = 𝑒 𝑖 𝜋 2 = 𝑖. Also replacing 𝜔 by -1 and 𝑻 by 𝒁 leads to an Eigenlogic operator for the three-input conjunction: 𝑪 𝑪𝒁 = 𝑮 𝑥∧𝑦∧𝑧 .

Interpolation Synthesis of Multivalued Quantum Gates

Multi-valued logic requires a different algebraic structure than ordinary binary-valued one. Many properties of binary logic do not support set of values that do not have cardinality 2 n . Multivalued logic is often used for the development of logical systems that are more expressive than Boolean systems for reasoning [START_REF] Miller | Multiple Valued Logic: Concepts and Representations[END_REF]. Particularly three and four valued systems, have been of interest with applications in digital circuits and computer science.

OAM-1 System, for Ternary Min and Max Logical Gates

The balanced logical system {+1.0, -1} has several benefits because it approaches the two logical systems most commonly used in binary logic {+1, -1} and {0, 1}, which are special cases of this ternary logic. Moreover, its values are centered in zero, thus assuring a simplification of the results and interesting properties of symmetry.

Orbital angular momentum (OAM) is characterized by two quantum numbers: ℓ the orbital number and m the magnetic number. The rules are: ℓ ≥ 0 is an integer and - ℓ ≤ 𝑚 𝑙 ≤ ℓ. The matrix form of the z-component orbital angular momentum observable for ℓ = 1 is:

𝑳 𝒛 = ℏ𝜦 = ℏ ( +1 0 0 0 0 0 0 0 -1 )
the three eigenvalues {+ 1, 0, - 1} are here considered the logical truth values.

One can now express the ternary logical observables as developments over the rank-1 projection operators spanning the vector space: 𝜫 +1 , 𝜫 0 and 𝜫 -1 . These operators are explicitly calculated by the interpolation formula and correspond to the density matrices of the three eigenstates | + 1 >, |0 > and | -1 > of 𝑳 𝒛 , defining a qutrit. The projection operators are function of the seed operator 𝜦 and are given by:

𝜫 +1 = 𝟏 𝟐 𝜦(𝜦 + 𝑰) , 𝜫 0 = 𝑰 -𝜦 𝟐 , 𝜫 -1 = 𝟏 𝟐 𝜦(𝜦 -𝑰)
All arity-one logical operators 𝑭(𝜦) can then be derived using the development over the projection operators.

When considering an arity-2 3-valued system, the operators are represented by 9 × 9 matrices. The dictators, 𝑼 and 𝑽, are then:

𝑼 = 𝜦 ⊗ 𝑰 , 𝑽 = 𝑰 ⊗ 𝜦 , 𝑼 ⋅ 𝑽 = 𝜦 ⊗ 𝜦
In ternary logic, popular connectives are 𝑀𝑖𝑛 and 𝑀𝑎𝑥 . Using the above found projection operators and logical reduction rules (due to the completeness of the projection operators) one obtains the following logical observables:

𝑴𝒊𝒏(𝑼, 𝑽) = 𝟏 𝟐 (𝑼 + 𝑽 + 𝑼 2 + 𝑽 2 -𝑼 ⋅ 𝑽 -𝑼 2 ⋅ 𝑽 2 ) = 𝑑𝑖𝑎𝑔(+1, +1, + 1, +1,0,0, +1,0, -1) 𝑴𝒂𝒙(𝑼, 𝑽) = 𝟏 𝟐 ( 𝑼 + 𝑽 -𝑼 2 -𝑽 2 + 𝑼 ⋅ 𝑽 + 𝑼 2 ⋅ 𝑽 2 ) = 𝑑𝑖𝑎𝑔(+1,0, -1,0,0, -1, -1, -1, -1)
The truth tables of the respective logical connective appear in the ordered eigenvalue system of the respective logical operator.

Quantum Gates for a Qutrit Balanced Calculator for Addition, Multiplication and Division

The adder is one of the fundamental components in digital electronics. One can, by the means of multivalued logic, improve the performances of this circuit by removing the delays caused by the propagation of the carry bit. Considering a balanced half-adder, the structure is simplified because the ternary logic values match the notation for balanced ternary digits.

The logic is here on the values {-1, 0, 1} (reversed from the preceding case) which have a natural correspondence to the numerical system. The seed operator is:

𝜦 𝑐 = ( -1 0 0 0 0 0 0 0 1 ) = 𝑑𝑖𝑎𝑔( - 1, 0, +1)
The projection operators are obtained by the same method as for the OAM case, giving here:

𝜫 -1 = 1 2 𝜦 𝑐 (𝜦 𝑐 -𝑰 3 ) , 𝜫 0 = 1 2 (𝑰 3 -𝜦 𝑐 2 ) , 𝜫 1 = 1 2 𝜦 𝑐 (𝜦 𝑐 + 𝑰 3 )
For an half-adder the inputs are represented by the addend operator 𝑨 = 𝜦 𝑐 ⊗ 𝑰 and the carryin operator 𝑪 𝑖 = 𝑰 ⊗ 𝜦 𝑐 .

The output operators are given by the sum 𝑺 𝑖 and the carry-out 𝑪 𝑖+1 . The corresponding logical observables are derived directly using the truth tables, where the truth values correspond to the diagonal elements of the corresponding logical observable, this gives the logical observables:

𝑺 𝑖 (𝑨, 𝑪 𝑖 ) = 𝑨 + 𝑪 𝑖 - 𝟑 𝟐 𝑨 2 ⋅ 𝑪 𝑖 - 𝟑 𝟐 𝑨 ⋅ 𝑪 𝒊 𝟐 = 𝑑𝑖𝑎𝑔(+1, -1,0, -1,0, +1,0, +1, -1)
𝑪 𝑖+1 (𝑨, 𝑪 𝑖 ) = 1 2 (𝑨 ⋅ 𝑪 𝒊 𝟐 + 𝑨 2 ⋅ 𝑪𝑖) = 𝑑𝑖𝑎𝑔(-1,0,0,0,0,0,0,0, +1)

The comparison of the truth tables in ternary logic [START_REF] Jones | Fast Ternary Addition[END_REF] with the ones used in binary systems shows that the sum function corresponds here to the modulo-3 sum function, and the carry function is the consensus function. Therefore, the implementation of a balanced ternary half-adder is natural. The half-adder will either increment or decrement, depending on whether 𝐶 0 is +1 or -1, while the binary equivalent can only increment.

Then one can naturally build the full-adder consisting of two addend inputs 𝐴 and 𝐵 and the carry-in 𝐶, The diagram is summarized on Figure 2.

In a more general way one can also derive, using the same method, the operators for multiplication and division. Every function 𝑭 of the operators 𝑨, 𝑩 and 𝑪 can be put in the developed form: 𝑭(𝑨, 𝑩, 𝑪) = ∑ 𝑓 𝑖𝑗𝑘 (𝑎, 𝑏, 𝑐) 𝜫 𝑖 ⊗ 𝜫 𝑗 ⊗ 𝜫 𝑘 1 𝑖,𝑗,𝑘=-1

For example considering the truth table for the carry-out for multiplication: 𝐶 𝑚𝑢𝑙𝑡 , one obtains directly the corresponding operator (which is a dimension 3 × 3 × 3 = 27 matrix):

𝑪 𝑚𝑢𝑙𝑡 = 𝑑𝑖𝑎𝑔(0,0, +1,0,0,0, -1,0,0,0,0,0,0,0,0,0,0,0, -1,0,0,0,0,0,0,0, +1) giving:

𝑪 𝑚𝑢𝑙𝑡 = (1)(𝜫 -1 ⊗ 𝜫 -1 ⊗ 𝜫 1 + 𝜫 1 ⊗ 𝜫 1 ⊗ 𝜫 1 ) + (-1)(𝜫 1 ⊗ 𝜫 -1 ⊗ 𝜫 -1 + 𝜫 -1 ⊗ 𝜫 1 ⊗ 𝜫 -1 )
The « input » logical operators are defined in the usual way in Eigenlogic as:

𝑨 = 𝜦 𝑐 ⊗ 𝑰 ⊗ 𝑰 = 𝑑𝑖𝑎𝑔(-1, -1, -1, -1, -1, -1, -1, -1, - 1
,0,0,0,0,0,0,0,0,0, +1, +1, +1, +1, +1, +1, +1, +1, +1) 𝑩 = 𝑰 ⊗ 𝜦 𝑐 ⊗ 𝑰 = 𝑑𝑖𝑎𝑔(-1, -1, -1,0,0,0, +1, +1, +1, -1, -1, -1,0,0,0, +1, +1, +1, -1, -1, -1,0,0,0, +1, +1, +1) 𝑪 = 𝑰 ⊗ 𝑰 ⊗ 𝜦 𝑐 = 𝑑𝑖𝑎𝑔(-1,0, +1, -1,0, +1, -1,0, +1, -1,0, +1, -1,0, +1, -1,0, +1, -1,0, +1, -1,0, +1, -1,0, +1)

The operators 𝑨 , 𝑩 and 𝑪 (named logical projectors or dictators for the three qutrit system) commute and correspond to the usual ordering (from left to right) in the truth tables. This leads to the following expression for the carry output for multiplication: = 𝑑𝑖𝑎𝑔(0,0, +1,0,0,0, -1,0,0,0,0,0,0,0,0,0,0,0, -1,0,0,0,0,0,0,0, +1)

𝑪 𝑚𝑢𝑙𝑡 = 1
which is relatively simple expression.

One can apply the same method (based on the respective truth table) in order to obtain the logical operators for 𝑺 and 𝑪 for the other arithmetical operations. This gives for the qutrit full-adder gate: = 𝑑𝑖𝑎𝑔(0, +1, -1, +1, -1,0, -1,0, +1, +1, -1,0, -1,0, +1,0, +1, -1, -1,0, +1,0, +1, -1, +1, -1,0) = 𝑑𝑖𝑎𝑔(-1, -1,0, -1,0,0,0,0,0, -1,0,0,0,0,0,0,0, +1,0,0,0,0,0, +1,0, +1, +1)

For the qutrit multiplier gate one obtains:

𝑺 𝑚𝑢𝑙𝑡 = 𝑨𝑩 + 𝑪 - 3 2 (𝑨 2 𝑩 2 𝑪 + 𝑨𝑩𝑪 2 ) =
= 𝑑𝑖𝑎𝑔(0, +1, -1, -1,0, +1, +1, -1,0, -1,0, +1, -1,0, +1, -1,0, +1, +1, -1,0, -1,0, +1,0, +1, -1)

𝑪 𝑚𝑢𝑙𝑡 = 1 2 (𝑨𝑩𝑪 2 + 𝑨 2 𝑩 2 𝑪) =
= 𝑑𝑖𝑎𝑔(0,0, +1,0,0,0, -1,0,0,0,0,0,0,0,0,0,0,0, -1,0,0,0,0,0,0,0, +1)

For the qutrit divider gate one obtains:

𝑺 𝑑𝑖𝑣 = 𝑨𝑩 + 𝑩 2 𝑪 - 3 2 (𝑨 2 𝑩 2 𝑪 + 𝑨𝑩𝑪 2 ) =
= 𝑑𝑖𝑎𝑔(0, +1, -1, -1, 0 ̅ , 0 ̅ , 0 ̅ , +1, -1,0, -1,0, +1, 0 ̅ , 0 ̅ , 0 ̅ , -1,0, +1, +1, -1,0, 0 ̅ , 0 ̅ , 0 ̅ , 0, +1, -

𝑪 𝑑𝑖𝑣 = 𝑪 𝑚𝑢𝑙𝑡 = 𝑑𝑖𝑎𝑔(0,0, +1, 0 ̅ , 0 ̅ , 0 ̅ , -1,0,0,0,0,0, 0 ̅ , 0 ̅ , 0 ̅ , 0,0,0, -1,0,0, 0 ̅ , 0 ̅ , 0 ̅ , 0,0, +1)

The symbol "0 ̅ " in the eigenvalue system signifies that for this value (which is here 0) the division 𝐴/𝐵 is not defined, this corresponds to the input 𝐵 = 0.

The expressions obtained here above show the symmetry between multiplication and division.

Discussion

A general method has been presented for the design of logical quantum gates. It uses matrix interpolation inspired from classical multivariate methods. The interesting property is that a unique seed operator generates the entire logical family of operators for a given 𝑚-valued 𝑛-arity system. When considering the binary alphabet {+1, -1} the gates are the quantum equivalent of the Fourier transform of Boolean functions. The method proposes a new expression of the Toffoli gate which can be understood as a 3-argument conjunction in the Eigenlogic interpretation. For multivalued logic quantum gates can be derived for different alphabets. The correspondence with quantum angular momentum leads to physical realizable gates. Applications have been presented for, Min-Max and qutrit arithmetic logical gates.

This opens a new perspective for quantum computation because several of the Eigenlogic operators turn out to be well-known quantum gates. This shows an operational correspondence between quantum control logic (Deutsch's paradigm [START_REF] Deutsch | Quantum theory, the Church-Turing principle and the universal quantum computer[END_REF]) and ordinary propositional logic. In Eigenlogic measurements on logical operators give the truth values of the corresponding logical connective.

In propositional logic the arguments of a compound logical proposition are considered the atomic propositions, in Eigenlogic, these are what we have named the logical projector operators. Examples are the one-argument logical projector 𝑨 and the two two-argument logical projectors 𝑷 1 and 𝑸 0 and the three argument logical projectors as defined in Section 3.2. This is a fundamental difference with what is usually considered in quantum logic where atomic propositions are associated with rays i.e., quantum pure state density matrices. In Eigenlogic the logical connective conjunction (𝐴𝑁𝐷, ∧), which is non-atomic, is the connective represented by a ray (rank-1 projection operator), the other rays are simply obtained by complementing selectively the arguments of the conjunction. From the point of view of logic atomic propositions must be independent propositions and this can only be achieved with the formulation given here using the Kronecker product as was done for 𝑷 1 and 𝑸 0 and not by mutually exclusive projection operators, such as the rank -1 projection operators which do not correspond to independent propositions. Thus for Eigenlogic, atomic propositions are not rays (or pure quantum state density matrices) when considering connectives with more than one argument (𝑛 ≥ 2).

At first sight the methods discussed here could be viewed as "classical" because of the identification of Eigenlogic with propositional logic. But because this approach uses quantum observables it can also be considered as being part of the global "quantum machinery". Most problems in traditional quantum physics deal with finding eigenfunctions and eigenvalues of some physical observable, the most investigated being the Hamiltonian observables whose eigenvalues represent the energies of a physical system and whose eigenstates are the stationary states representing the stable equilibrium solutions, in the form of wavefunctions, of the Schrödinger equation. The non-traditional aspects of quantum mechanics, principally superposition, entanglement and non-commutativity, are largely employed in the field of quantum information and are considered as a resource for quantum computing. Considering quantum states, that are not eigenvectors, these could be entangled states, the measurement outcomes are governed by the probabilistic quantum Born rule, and interpretable results are then the mean values, this leads to a fuzzy logic.

  where | 𝜑 𝑝 ≥ 𝑐𝑜𝑠 𝜃 𝑝 2 |0 > + 𝑒 𝑖𝜙 𝑝 𝑠𝑖𝑛 𝜃 𝑝 2 |1 > (for | 𝜑 𝑞 > we have a similar expression). Now putting 𝑠𝑖𝑛 𝜃 𝑝 2 = 𝑝 and 𝑠𝑖𝑛 𝜃 𝑞 2 = 𝑞 represent the probabilities of being in the "True" state |1 > for spins 1/2 oriented along two different axes 𝜃 𝑝 and 𝜃 𝑞 .

Figure 1 .

 1 Figure 1. Bloch sphere with the qubit state |𝜙⟩ characterized by the angles 𝜃 and 𝜑.

  of 𝑚 + 1 values as used in Łukasiewicz logic. Other convenient numerical choices are i.e. the balanced ternary values {+1,0, -1} for qutrits.When the eigenvalues are real the corresponding logical operators are observables, e.g., Hermitian operators. Many logical gates are of this type, the most popular ones are the one-qubit , Pauli-𝑿, Hadamard 𝑯, phase 𝑺 and 𝑻 gates, the two-qubit controlled-not, controlled-𝒁 gate and the swap gate and the universal three-qubit Toffoli gate 𝑻𝑶.

  1) , 𝑭 𝐴𝑁𝐷 |𝑥𝑦 > = 𝑥𝑦 |𝑥𝑦 > applying this operator to the state |𝑥𝑦 > one obtains the eigenvalue 𝑥𝑦 = 1, True, only for the state |11 >. One can transform this operator using the Householder transform in: 𝑮 𝐴𝑁𝐷 = 𝑰 -2𝑭 𝐴𝑁𝐷 = ( - 1) 𝜫⊗𝜫 = 𝑑𝑖𝑎𝑔 𝑧 (1, 1, 1, - 1) = 𝑪 𝒁 𝑮 𝐴𝑁𝐷 |𝑥𝑦 >= 𝑪 𝒁 |𝑥𝑦 > = ( - 1) 𝑥𝑦 |𝑥𝑦 > justifying the interpretation of the 𝑪 𝒁 gate as a conjunction in Eigenlogic for the alphabet {+ 1, - 1}.

Figure 2 .

 2 Figure 2. General arithmetic operation diagram in a qu-trit system.

Table 1 .

 1 The sixteen two-argument logical observables and their truth-tables.

	Connective 𝑨,

𝑩 Truth Table {F,T} {𝟎

			, 𝟏} Projective	{ + 𝟏, - 𝟏} Isometric
	𝐹𝑎𝑙𝑠𝑒 ; F ; ⊥	𝐹 𝐹 𝐹 𝐹	0	+ 𝑰
	𝑁𝑂𝑅	𝐹 𝐹 𝐹 𝑇	𝑰 -𝑨 -𝑩 + 𝑨𝑩	(1/2) (+𝑰 -𝑼 -𝑽 -𝑼𝑽)
	𝐴 ⇍ 𝐵	𝐹 𝐹 𝑇 𝐹	𝑩 -𝑨𝑩	(1/2) (+𝑰 -𝑼 + 𝑽 + 𝑼𝑽)
	¬𝐴	𝐹 𝐹 𝑇 𝑇	𝑰 -𝑨	- U
	𝐴 ⇏ 𝐵	𝐹 𝑇 𝐹 𝐹	𝑨 -𝑨𝑩	(1/2) ( + 𝑰 + 𝑼 -𝑽 + 𝑼𝑽)
	¬𝐵	𝐹 𝑇 𝐹 𝑇	𝑰 -𝑩	- 𝑽
	𝑋𝑂𝑅 ; 𝐴 ⊕ 𝐵	𝐹 𝑇 𝑇 𝐹	𝑨 + 𝑩 -2 𝑨𝑩	𝑼𝑽 = 𝒁 ⊗ 𝒁
	𝑁𝐴𝑁𝐷 ; 𝐴 ↑ 𝐵	𝐹 𝑇 𝑇 𝑇	𝑰 -𝑨𝑩	(1/2) (- 𝑰 -𝑼 -𝑽 + 𝑼𝑽)
	𝐴𝑁𝐷 ; 𝐴 ∧ 𝐵	𝑇 𝐹 𝐹 𝐹	𝑨𝑩 = 𝜫 ⊗ 𝜫	(1/2) ( + 𝑰 + 𝑼 + 𝑽 -𝑼𝑽)
	𝐴 ≡ 𝐵	𝑇 𝐹 𝐹 𝑇	𝑰 -𝑨 -𝑩 + 2 𝑨𝑩	- 𝑼𝑽
	𝐵	𝑇 𝐹 𝑇 𝐹	𝑩 = 𝑰 ⊗ 𝜫	𝑽 = 𝑰 ⊗ 𝒁
	𝐴 ⇒ 𝐵	𝑇 𝐹 𝑇 𝑇	𝑰 -𝑨 + 𝑨𝑩	(1/2) ( - 𝑰 -𝑼 + 𝑽 -𝑼𝑽)
	𝐴	𝑇 𝑇 𝐹 𝐹	𝑨 = 𝜫 ⊗ 𝑰	𝑼 = 𝒁 ⊗ 𝑰
	𝐴 ⇐ 𝐵	𝑇 𝑇 𝐹 𝑇	𝑰 -𝑩 + 𝑨𝑩	(1/2) (-𝑰 + 𝑼 -𝑽 -𝑼𝑽)
	𝑂𝑅 ; 𝐴 ∨ 𝐵	𝑇 𝑇 𝑇 𝐹	𝑨 + 𝑩 -𝑨𝑩	(1/2) ( -𝑰 + 𝑼 + 𝑽 + 𝑼𝑽)
	𝑇𝑟𝑢𝑒 ; T	𝑇 𝑇 𝑇 𝑇	𝑰	- 𝑰

Acknowledgments:

We would like to thank Benoît Valiron from CentraleSupélec and LRI (Laboratoire de Recherche en Informatique), Gif-sur-Yvette (FR) for fruitful discussions during an ongoing academic project on quantum programming and for having pointed out the work using T gates. We are also very grateful to Francesco Galofaro, from Politecnico Milano (IT) and Free University of Bolzano (IT) for his pertinent advices on semantics and logic. We would like to thank the students of CentraleSupelec who worked on the multivalued quantum gates: Noémie Perrot, Mathieu Ha-Sum, Alexandre Criseo and Jorge Fernandez-Mayoralas.


T Chauve

M Montagnat

S Piazolo

B Journaux

J Wheeler

F Barou

D Mainprice

A Tommasi

Non-basal dislocations should be accounted for in simulating ice mass flow

Keywords: Non-basal dislocations in ice, Weighted Burgers Vectors, cryo-EBSD, crystal plasticity

Prediction of ice mass flow and associated dynamics is pivotal at a time of climate change. Ice flow is dominantly accommodated by the motion of crystal defects -the dislocations. In the specific case of ice, their observation is not always accessible by means of the classical tools such as X-ray diffraction or transmission electron microscopy (TEM). Part of the dislocation population, the geometrically necessary dislocations (GNDs) can nevertheless be constrained using crystal orientation measurements via electron backscattering diffraction (EBSD) associated with appropriate analyses based on the Nye (1950) approach. The present study uses the Weighted Burgers Vectors, a reduced formulation of the Nye theory that enables the characterization of GNDs. Applied to ice, this method documents, for the first time, the presence of dislocations with non-basal [c] or < c+a > Burgers vectors. These [c] or < c+a > dislocations represent up to 35% of the GNDs observed in laboratory-deformed ice samples.

Our findings offer a more complex and comprehensive picture of the key plasticity processes responsible for polycrystalline ice creep and provide better constraints on the constitutive mechanical laws implemented in ice sheet flow models used to predict the response of Earth ice masses to climate change.

Introduction

Understanding the deformation behavior of ice crystals is essential for modeling the flow of glaciers and ice sheets. Ice on Earth, ice Ih, has an hexagonal crystalline structure. It has a strong viscoplastic anisotropy, since deformation occurs almost exclusively by dislocation glide on the basal plane [START_REF] Duval | Rate controlling processes in the creep of polycrystalline ice[END_REF]. This crystal-scale anisotropy results in strong textures (crystallographic orientations) and, hence, in large-scale texture-induced anisotropy.

This anisotropy has crucial effects on large-scale ice flow (e.g. [START_REF] Durand | Change of the ice rheology with climatic transitions. Implication on ice flow modelling and dating of the EPICA Dome C core[END_REF]). It is responsible, for instance, for abrupt changes in rheology between the ice sheet and the ice shelf [START_REF] Ma | Enhancement factors for grounded ice and ice shelves inferred from an anisotropic ice-flow model[END_REF] and for basal folding [START_REF] Bons | Converging flow and anisotropy cause large-scale folding in Greenland's ice sheet[END_REF]. The viscoplastic anisotropy of ice crystals also results in strong strain and stress heterogeneity [START_REF] Grennerat | Experimental characterization of the intragranular strain field in columnar ice during transient creep[END_REF], leading to dynamic recrystallization [START_REF] Duval | Rate controlling processes in the creep of polycrystalline ice[END_REF][START_REF] Chauve | Strain field evolution during dynamic recrystallization nucleation; a case study on ice[END_REF], a process that is essentially controlled by the dislocation behavior and interactions [START_REF] Chauve | Investigation of nucleation processes during dynamic recrystallization of ice using cryo-EBSD[END_REF].

Ice is therefore a good analogue to study the behaviour of materials with high viscoplastic anisotropy deforming at high temperature (T /T melt > 0.9), such as the Earth lower crust and mantle, where the dominant rock-forming minerals (e.g. feldspar, quartz, olivine, pyroxenes, micas) are highly anisotropic.

However, the difficulty in observing dislocations by TEM or X-ray diffraction results in a lack of knowledge on the activity of other slip systems or of mechanisms such as climb or cross-slip that may complement basal glide. The lack of constraints on the activity of the non-basal slip systems in ice limits the ability of micro-macro crystal plasticity methods to simulate the mechanical behaviour of ice and its evolution (see [START_REF] Montagnat | Multiscale modeling of ice deformation behavior[END_REF] for a review). To approach a realistic mechanical behaviour, which can be used to model the flow of glaciers and polar ice sheets, strong assumptions have been made (see [START_REF] Castelnau | Modelling viscoplastic behavior of anisotropic polycrystalline ice with a self-consistent approach[END_REF]; [START_REF] Kennedy | The evolution of crystal fabric in ice sheets and its link to climate history[END_REF] for instance). In particular, in all models based on crystal plasticity, four to five independent slip systems are required to maintain strain compatibility [START_REF] Hutchinson | Creep and plasticity of hexagonal polycrystals as related to single crystal slip[END_REF], hence for ice, glide on non-basal slip systems is allowed. [START_REF] Castelnau | Modelling viscoplastic behavior of anisotropic polycrystalline ice with a self-consistent approach[END_REF] imposed a non basal activity 70 times harder than basal activity, while [START_REF] Llorens | Full-field predictions of ice dynamic recrystallisation under simple shear conditions[END_REF] lowered this ratio to 20, enabling a significant contribution of non-basal systems to deformation, without any experimental evidence to stand on.

Most observed dislocations in ice so far have one of the three equivalent 1/3 < 2 11 0 > Burgers vectors and are constrained to glide in the basal plane (0001) owing to their tendency to dissociate into partial dislocations [START_REF] Higashi | Lattice defects in ice crystals[END_REF][START_REF] Hondoh | Nature and behavior of dislocations in ice[END_REF]. Rare 1/3 < 2 11 0 > dislocations have been observed to glide on prismatic planes by X-ray diffraction in low strain conditions where very few dislocations were activated [START_REF] Shearwood | X-ray topographic observations of edge dislocation glide on non-basal planes in ice[END_REF], and when crystals were oriented to minimize the resolved shear stress in the basal plane [START_REF] Liu | Thermally induced dislocation loops in polycrystalline ice[END_REF]. Indirect evidence of double cross-slip of basal dislocations was obtained from X-ray diffraction observations on single crystal deformed in torsion [START_REF] Montagnat | The heterogeneous nature of slip in ice single crystals deformed under torsion[END_REF]. Dislocation Dynamic simulations estimated the local stress necessary to activate this mechanism [START_REF] Chevy | Creep behaviour of ice single crystals loaded in torsion explained by dislocation cross-slip[END_REF].

So far, direct observations (via X-ray diffraction) of dislocations with Burgers vector [c] =

[0001] or < c + a >= 1/3 < 11 23 > are limited to very specific conditions such as peripheral dislocations of stacking faults formed during crystal growth or under cooling [START_REF] Higashi | Lattice defects in ice crystals[END_REF].

The formation of stacking faults under cooling is assumed to result from climb of the basal component of dislocation loops with < c + a > Burgers vector, induced by the precipitation of excess point defects generated by cooling. Dislocation loops with [c]-component Burgers vectors were also observed to form due to tiny inclusions (water droplets for pure ice, or solute pockets for NH 3 -doped ice) formed during crystallization and due to thermal stress imposed in the crystal growing apparatus [START_REF] Oguro | Concentric dislocation loops with [0001] Burgers vectors in ice single crystals doped with NH 3[END_REF]). To our knowledge, there are no other direct observations of dislocations with a [c]-component Burgers vector for pure or natural ice. Weikusat et al. (2011b) indirectly inferred [c] or < c + a > dislocations as necessary to explain some subgrain boundary structures observed in ice core samples. The techniques used (surface sublimation to extract subgrain boundaries and discrete X-ray Laue diffraction analyses to obtain local orientations along profiles) did not provide full constraints on the nature of the subgrain boundaries. Nevertheless, by assuming that the subgrain boundaries were perpendicular to the surface, some could be interpreted as tilt boundaries composed of [c] or

< c + a > dislocations.
Dislocations are nucleated and contribute to plastic deformation by gliding. The dislocations can be stored in the microstructure by two modes; as trapped dislocations due to dislocation interaction, called Statistically Stored Dislocations (SSDs) and as Geometrically Necessary Dislocations (GNDs) [START_REF] Fleck | Strain gradient plasticity : theory and experiment[END_REF]. GNDs are intimately associated with lattice curvature, and hence contribute to local strain that can be detected by EBSD as misorientation gradients. They contribute to heterogeneous plastic strain, such as bending or twisting but they can develop even though the experimental conditions allows the possibility of a homogeneous deformation (Van der [START_REF] Van Der Giessen | GNDs in nonlocal plasticity theories: lessons from discrete dislocation simulations[END_REF]. It is generally acknowledged that density of GNDs is significantly higher than density of SSDs [START_REF] Kubin | Geometrically necessary dislocations and strain-gradient plasticity: a few critical issues[END_REF].

EBSD analyses of ice were recently made possible thanks to cryo-stages able to maintain samples at very cold temperatures (-100 to -150 • C), under low vacuum. This technique gives access to full crystal orientations over reasonably large polycrystalline samples (few cm 2 ), with a good spatial resolution (down to 0.1 µm). The first applications of EBSD on ice were oriented towards full crystal orientation measurements at the grain level [START_REF] Obbard | Using electron backscatter diffraction patterns to examine recrystallization in polar ice sheets[END_REF]. High spatial resolution crystal misorientations within grains were recently used to characterize dislocation substructures [START_REF] Piazolo | Sub-structure characterization of experimentally and naturally deformed ice using cryo-EBSD[END_REF][START_REF] Montagnat | Measurements and full-field predictions of deformation heterogeneities in ice[END_REF]Weikusat et al., 2011a;[START_REF] Montagnat | Analysis of dynamic recrystallization of ice from EBSD orientation mapping[END_REF][START_REF] Chauve | Investigation of nucleation processes during dynamic recrystallization of ice using cryo-EBSD[END_REF]. EBSD observations performed in the above mentioned studies are post-mortem and therefore record the effects of the GNDs remaining after relaxation of the internal stress field through anelastic deformation.

Since conventional EBSD maps are 2D, they do not give access to the full dislocation (Nye) tensor α but only to five components (α 12 , α 21 , α 13 , α 23 , α 33 ) where the subscript 3 refers to the normal to the EBSD surface. By this mean, EBSD observations provide lower bounds of GND density [START_REF] Pantleon | Resolving the geometrically necessary dislocation content by conventional electron backscattering diffraction[END_REF]. Recently, Wheeler et al. (2009) proposed a method of characterization of the GNDs called the "weighted Burgers vector" (WBV) (see Appendix A for a detailed description). It corresponds to the projection of the Nye tensor on the EBSD surface and can be expressed as W BV = (α 13 , α 23 , α 33 ). The WBV tool does not aim at approaching the full dislocation density tensor (as attempted by [START_REF] Pantleon | Resolving the geometrically necessary dislocation content by conventional electron backscattering diffraction[END_REF] for instance), but does not require the third dimension to provide meaningful information about the GND population. Its amplitude gives a lower bound for the density of GNDs and its direction refers to the Burgers vector of the sampled GNDs. One important point is that although the WBV does not record all the GNDs present, it cannot contain phantom directions. If it has a significant [c]-component then at least some of the Burgers vectors of the GNDs must have a [c]-component though this does not mean they have to be parallel to [c].

As for the Nye tensor, the WBV analysis only reflects the GND contribution to the dislocation density. Without further assumptions, this contribution cannot be directly related to the mobile dislocations responsible for most of the plastic deformation.

Cryo-EBSD associated with the WBV analysis was recently shown to be very efficient to characterize the nature of GNDs in ice [START_REF] Piazolo | Effect of local stress heterogeneities on dislocation fields: Examples from transient creep in polycrystalline ice[END_REF]. Although restricted to small areas, this previous study revealed a contribution of dislocations with [c]-or < c + a >component. These observations encouraged us to perform new EBSD observations on ice polycrystals deformed in the laboratory, with a higher spatial resolution and over larger areas than in the preliminary study of [START_REF] Piazolo | Effect of local stress heterogeneities on dislocation fields: Examples from transient creep in polycrystalline ice[END_REF].

The present work aims therefore at (i) documenting the presence of dislocations with Burgers vectors comprising a component along [c], (ii) estimating quantitatively the significance of these dislocations within the observed GNDs, and (iii) discussing the implication of this observation for the micro-macro modeling of ice mechanical behaviour, up to the scale of glaciers and polar ice flow.

Material and Methods

Large ice polycrystalline samples were deformed in torsion and uniaxial unconfined compression under constant imposed load at high homologous temperature (T /T melt ∼ 0.98, in a cold room). The samples deformed by compression had a columnar initial grain shape with large grain size (1 to 4 cm 2 ) (see [START_REF] Grennerat | Experimental characterization of the intragranular strain field in columnar ice during transient creep[END_REF]; [START_REF] Chauve | Investigation of nucleation processes during dynamic recrystallization of ice using cryo-EBSD[END_REF] for details) and were deformed under a constant load of 0.5 M P a applied in the plane perpendicular to the column directions, up to a macroscopic strain of about 3%. Torsion tests were performed on solid cylinders (radius × height = 18 mm × 60 mm) of granular ice (millimetre grain size), under a maximum applied shear stress at the outer radius between 0.5 and 0.6 M P a (experimental conditions similar to the ones in [START_REF] Bouchez | The fabric of polycrystalline ice deformed in simple shear : experiments in torsion, natural deformation and geometrical interpretation[END_REF]; [START_REF] Montagnat | The heterogeneous nature of slip in ice single crystals deformed under torsion[END_REF]). Several tests enabled to cover a range of maximum shear strain between 0.01 and 2. These two experimental conditions are complementary. The compression tests enable to follow the first step of deformation in a model microstructure invariant in the third dimension (parallel to the columns) that is close to a 2.5D configuration, where surface observations are a good proxy to the bulk mechanisms [START_REF] Grennerat | Experimental characterization of the intragranular strain field in columnar ice during transient creep[END_REF]. The torsion experiments give access to large strain levels on an initially isotropic microstructure and texture. A summary of the experimental conditions of the tests used in this study is given in table 1.

Id

Sample Mechanical test T Samples (20×10×3 mm 3 ) were extracted from the deformed blocks for cryo-EBSD observations (angular resolution of 0.7 • , spatial resolution of 5 and 20 µm for this study). The torsion samples were cut perpendicular to the radius, as close as possible to the external side of the cylinder. Appropriate adjustment of the vacuum and temperature (1 Pa and -100 • C) to reduce sublimation was made following [START_REF] Montagnat | Analysis of dynamic recrystallization of ice from EBSD orientation mapping[END_REF]. This allowed EBSD mapping of the entire selected areas with indexation rates higher than 85%.

At the compression and shear strains reached, dynamic recrystallization mechanisms such as nucleation at triple junctions, highly misoriented subgrain boundaries and kink bands are observed as in [START_REF] Chauve | Investigation of nucleation processes during dynamic recrystallization of ice using cryo-EBSD[END_REF]. In the compression sample analyses we focused on grain boundaries and triple junction areas. "Typical" dislocation substructures are presented in figures 1 and 2. Similar features were observed in the torsion samples. In these samples, we analyzed larger areas in order to obtain statistical information about the nature of the dislocations involved in the observed substructures.

In order to characterize the dislocations involved in the formation of subgrain boundaries, The WBV analysis gives a vector which can be expressed in the crystal or sample reference frame.

The WBV analyses were performed as (i) a point by point analysis that enables to plot the WBV direction and magnitude along the dislocation substructures and (ii) an integral WBV calculation of the net Burgers vector content of dislocations intersecting a given area of a map by an integration around the edge of this area. The integral WBV calculated over a given area is projected over the four non-independent lattice components of the hexago-

nal symmetry ([11 20],[ 2110],[1 210] and [0001] noted W BV a1 , W BV a2 , W BV a3 , W BV c ).
This integral WBV analysis complements the point-by-point WBV calculations and, due to the integration over an area, reduces the noise level in the analysis (see Appendix A). In special cases the integration also induces a loss of information. For instance, in the case of an integral calculation over an area containing a perfect kink band, the resulting integral WBV will be null if the two opposite tilt bands have similar misorientation angles.

The proportion of dislocations with a [c]-component Burgers vector (that includes dislocations with [c] and < c + a > Burgers vectors, thereafter referred to as [c]-component dislocations)

in the subgrain boundaries is estimated as the ratio between the W BV c component over the Euclidian norm of the WBV (|W BV c |/||WBV||), thereafter called rW BV c . For the pixel-scale calculations, a cut-off value was defined in such a way to restrict the analysis to sub-structures with a misorientation higher than 0.9 • , to remain slightly above the EBSD resolution. This cut-off value transposed to the WBV norm depends on the EBSD step-size since the WBV is calculated per unit length (1.4 × 10 -3 µm -1 for 5 µm EBSD step size and 3.5 × 10 -4 µm -1 for 20 µm EBSD step size). Subgrain boundaries are distinguished by selecting the pixels for which the norm of the WBV is higher than this threshold and lower that the upper bound for a subgrain boundary set at 7 • of misorientation [START_REF] Chauve | Investigation of nucleation processes during dynamic recrystallization of ice using cryo-EBSD[END_REF]. The cut-off value is coherent with the limit of accuracy of EBSD data and leads to a good agreement with the subgrain boundary segmentation defined based on the local misorientation only. By doing so, less that 1% of pixels are selected as "sub-structures" in the non deformed sample and the corresponding values of |W BV c |/||WBV|| are uniformly distributed.

The WBV analyses are associated with classical measurements of the rotation axis of the misorientation induced by the subgrain boundary (by making use of absolute orientations from EBSD data) together with the orientation of the boundary trace. From this method known as "boundary trace analysis" [START_REF] Mainprice | Individual orientation measurements in quartz polycrystals: advantages and limitations for texture and petrophysical property determinations[END_REF][START_REF] Lloyd | Misorientation analysis and the formation and orientation of subgrain and grain boundaries[END_REF][START_REF] Prior | Some garnet microstructures: an illustration of the potential of orientation maps and misorientation analysis in microstructural studies[END_REF][START_REF] Piazolo | Sub-structure characterization of experimentally and naturally deformed ice using cryo-EBSD[END_REF], the boundary plane can be inferred. These information are used as a visualisation tool in figure 2.

Finally, statistical analyses were performed by using a probability density function that represents the ratio between the number of pixels with a WBV norm higher than the threshold (defined above) over the total number of pixels. It can be seen, for instance in figure 4, that this ratio is small for the low torsion strain experiment. The pixels with a WBV norm higher than the threshold are also separated as a function of the nature of the WBV, meaning mostly composed of [c]-component dislocations, mostly composed of < a > dislocations, or composed of a similar amount of both types of dislocations.

Experimental observations

We present first detailed observations of a few subgrain boundaries that illustrate the techniques used to distinguish < a > from [c]-component Burgers vectors on GND substructures, and then a global analysis performed over large-scale EBSD maps containing hundreds of grains (from the torsion test samples), which aims at evaluating the statistical significance of the dislocations with [c]-component Burgers vectors within the substructures.

Frequently observed subgrain structures in ice deformed by plasticity include "closed" shaped subgrain boundaries (SGBs) formed in the vicinity of serrated grain boundaries (Fig. 1), in areas where the microstructure is very heterogeneous. These "closed" shaped SGBs were shown in [START_REF] Chauve | Investigation of nucleation processes during dynamic recrystallization of ice using cryo-EBSD[END_REF] The "closed loop" substructure on the left side of figure 1 has been selected for a detailed characterization (Fig. 2 and Table 2). It can be separated into three domains with distinct WBV orientations. Two of them, domains 1 and 3, have WBV orientations pointing in two opposite < m > (< 1 100 >) axis directions. These two subgrains accommodate a rotation around an axis parallel to the boundary plane (along < a > axes) but with opposite rotation directions. Such a configuration, characteristic of two tilt-bands with opposite signs, forming a kink band, is frequently observed in ice [START_REF] Montagnat | Measurements and full-field predictions of deformation heterogeneities in ice[END_REF][START_REF] Piazolo | Effect of local stress heterogeneities on dislocation fields: Examples from transient creep in polycrystalline ice[END_REF].

The subgrain boundary in domain 2 is characterized by a boundary plane that is perpendicular to the ones of the SBGs from domains 1 and 3. However its rotation axis is also parallel to an < a > axis and it is contained within the subgrain boundary plane. Subgrain

Analyzed

Integral WBV µm to 1.94, for which EBSD observations reveal a high density of subgrain boundaries (Fig. 3). For each sample, the local WBV analysis was performed over the entire mapped surface (about 20×10 mm 2 ). In figure 3a component Burgers vector does not seem to be correlated with the orientation of the crystal.

-2 area W BV a1 W BV a2 W BV a3 W BV c |W BV c |/||WBV|| 1 -2.
To further test this point, orientation data at the pixel scale were correlated with the relative amplitude of the WBV components. To do so, we selected data from the sample deformed by torsion at γ = 0.42 (TGI03), since at this rather low shear strain the macroscopic texture remains reasonably weak to provide a wide enough orientation range (Fig. 5).

As performed in [START_REF] Grennerat | Experimental characterization of the intragranular strain field in columnar ice during transient creep[END_REF], an adapted Schmid factor, that does not account for slip direction, is used to describe the pixel orientation relative to the imposed stress configuration (S = |σ.c| 2 -(c.σ.c) 2 , where σ is the stress tensor and c is the c-axis orientation).

The distribution of this Schmid factor (Fig. 5) reveals a slight under representation of orientation with low Schmid factors, which may slightly bias the statistics. With this limitation in mind, figure 5 gives an overview of the relative contributions of the different components of the WBV as a function of the Schmid factor, and therefore as a function of the orientation of the pixel. First, the density of substructures (evaluated by the density of pixels with a WBV norm higher than the threshold) is similar independently of the crystallographic orientation.

The slight increase with Schmid factor must result from a statistical bias due to different number of pixels analysed for each orientation range (see top of Fig. 5). Second, dislocations with a [c]-component occur within similar proportions for every orientation. This statistical analysis confirms that there is no clear relationship either between local orientation and the density of GNDs, or between local orientation and the type of dislocations involved in the GND substructures.

Discussion

From these results, one important observation can be emphasized. Dislocations in ice, more specifically here GNDs, are clearly not composed solely by dislocations with < a > Burgers vectors. A non negligible amount of dislocations with a [c] component in their Burgers vectors contributes to the formation of subgrain boundaries in various configurations (boundary conditions, strain levels...) under laboratory conditions. Dislocations with a [c] component Burgers vector are theoretically energetically unfavourable, and possess a Peierls barrier up to 10 times the one of < a > dislocations [START_REF] Hondoh | Nature and behavior of dislocations in ice[END_REF].

They require therefore a higher level of resolved shear stress to be activated. Previous work on ice highlighted the link between local subgrain boundary development and local strain and/or stress concentrations based on misorientation measurements associated with full-field modeling approach [START_REF] Montagnat | Measurements and full-field predictions of deformation heterogeneities in ice[END_REF][START_REF] Piazolo | Effect of local stress heterogeneities on dislocation fields: Examples from transient creep in polycrystalline ice[END_REF], on direct comparison between strain field estimation by Digital Image Correlation and microstructure observations [START_REF] Chauve | Strain field evolution during dynamic recrystallization nucleation; a case study on ice[END_REF] and full-field modeling predictions [START_REF] Grennerat | Experimental characterization of the intragranular strain field in columnar ice during transient creep[END_REF]. Based on these recent works, we can assume that the combined effect of local redistribution of stress due to strain incompatibilities between grains [START_REF] Duval | Rate controlling processes in the creep of polycrystalline ice[END_REF][START_REF] Montagnat | Measurements and full-field predictions of deformation heterogeneities in ice[END_REF][START_REF] Piazolo | Effect of local stress heterogeneities on dislocation fields: Examples from transient creep in polycrystalline ice[END_REF] and the built up of dislocation fields and their associated internal stress field [START_REF] Chevy | Creep behaviour of ice single crystals loaded in torsion explained by dislocation cross-slip[END_REF][START_REF] Richeton | Modelling the transport of geometrically necessary dislocations on slip systems: application to singleand multi-crystals of ice[END_REF] may produce local stresses that allow the activation of non-basal slip systems or the glide of non-basal dislocations, and in particular [c]-component dislocations as observed here. The assumed link between local stress concentrations and formation of GNDs is consistent with high-resolution EBSD measurements recently performed on copper which show a correlation between high GND density and high intragranular residual stresses, directly inferred from HR-EBSD [START_REF] Jiang | Evolution of intragranular stresses and dislocation densities during cyclic deformation of polycrystalline copper[END_REF].

WBVs only capture part of the GNDs, which are, in turn, a fraction of the total dislocation population. The total contribution of [c]-component dislocations may therefore differ from the present estimations. Moreover, the GNDs populations observed on post-morten 2D cryo-EBSD data might not be proportional to the population of glissile dislocations, that is, representative of the relative activity of the slip systems, which are responsible for deformation. However they are responsible for the accommodation of stress heterogeneities through their contribution to the formation of the subgrain boundaries. By the association of strain measurements by DIC and microstructural observations, [START_REF] Chauve | Strain field evolution during dynamic recrystallization nucleation; a case study on ice[END_REF] demonstrated that the formation of subgrain boundaries lead to a marked strain redistribution within a polycrystal, which in the case of kink bands resulted in shear along the newly formed boundaries. GNDs act significantly during dynamic recrystallization by controlling nucleation by SIBM for instance [START_REF] Piazolo | Effect of local stress heterogeneities on dislocation fields: Examples from transient creep in polycrystalline ice[END_REF][START_REF] Chauve | Strain field evolution during dynamic recrystallization nucleation; a case study on ice[END_REF]. Last but not least, the internal stress fields resulting from dislocation fields [START_REF] Varadhan | Coupling the dynamic of statistically distributed and excess dislocations[END_REF] can induce the activation of mechanisms such as climb and cross-slip [START_REF] Montagnat | The heterogeneous nature of slip in ice single crystals deformed under torsion[END_REF].. We therefore expect GNDs to play a significant role during deformation at local and large scales.

All micro-macro modelling approaches applied to ice are so far based on drastic assumptions concerning the activated slip systems and on the mechanisms accommodating strain.

These assumptions are directly projected on the plasticity (or visco-plasticity) laws describing the dislocation glide and interactions during deformation. In most homogenization approaches (mean or full-field), plasticity is assumed to occur only through dislocation slip on at least four independent slip systems, and their interactions are taken into account by the critical resolved shear stresses which control the relative activities of the various slip systems, and their evolution laws. These laws are generally adjusted based on comparison of the modelled macroscopic mechanical response with experimental results [START_REF] Castelnau | Modelling fabric development along the GRIP ice core, central Greenland[END_REF][START_REF] Castelnau | Elastoviscoplastic micromechanical modeling of the transient creep of ice[END_REF][START_REF] Suquet | Multi-scale modeling of the mechanical behavior of polycrystalline ice under transient creep[END_REF]. These assumptions led to an unavoidable minimal activation of non-basal slip which compensates for the lack of knowledge of accommodating mechanisms (climb and cross-slip for instance), and the inability of the models to represent them, except for a few attempts [START_REF] Lebensohn | Modeling the mechanical response of polycrystals deforming by climb and glide[END_REF][START_REF] Lebensohn | Improved constitutive description of single crystal viscoplastic deformation by dislocation climb[END_REF]. The non-basal activity, and more specifically the fact that a minimum of pyramidal slip-system activity associated with [c] dislocations is always necessary was, until now, not justified by any observations. Most of these empirically adjusted parameters were used in further applications with, sometimes, limited validation tests [START_REF] Lebensohn | Modeling viscoplastic behavior and heterogeneous intracrystalline deformation of columnar ice polycrystals[END_REF][START_REF] Montagnat | Measurements and full-field predictions of deformation heterogeneities in ice[END_REF][START_REF] Grennerat | Experimental characterization of the intragranular strain field in columnar ice during transient creep[END_REF][START_REF] Llorens | Full-field predictions of ice dynamic recrystallisation under simple shear conditions[END_REF]. [START_REF] Devincre | Dislocation Mean Free Paths and Strain Hardening of Crystals[END_REF] or Dislocation Field approaches [START_REF] Taupin | Effects of size on the dynamics of dislocations in ice single crystals[END_REF][START_REF] Taupin | Rearrangement of dislocation structures in the aging of ice single crystals[END_REF][START_REF] Richeton | Modelling the transport of geometrically necessary dislocations on slip systems: application to singleand multi-crystals of ice[END_REF] and the actual distribution of < a > and [c]component dislocations in experimentally and naturally deformed ice samples as performed in [START_REF] Richeton | Modelling the transport of geometrically necessary dislocations on slip systems: application to singleand multi-crystals of ice[END_REF]. Finally, it suggests the necessity to introduce secondary mechanisms such as climb and cross-slip in the micro-macro approaches just mentioned, that is, of simulating the complexity of dislocation interactions and assessing its impact on the mechanical behaviour.

Only recently attempts have been made to consider the long-range internal stress field associated with dislocation substructures in crystal plasticity models [START_REF] Taupin | Effects of size on the dynamics of dislocations in ice single crystals[END_REF][START_REF] Taupin | Rearrangement of dislocation structures in the aging of ice single crystals[END_REF][START_REF] Richeton | Modelling the transport of geometrically necessary dislocations on slip systems: application to singleand multi-crystals of ice[END_REF]. These approaches, based on the elastic theory of continuously distributed dislocations account for the build up of GNDs and their transport during plasticity but they are limited to multi-crystals with few grains (∼ 20) because of numerical costs.

The validation of these approaches could strongly benefit from an accurate description of the nature of GNDs such as the one presented here. They could, in turn, provide constraints on the internal stress field favorable for the activation of [c]-component dislocations.

Coupling detailed analyses of dislocation substructures, like the one presented here, with such models, will produce a new generation of crystal plasticity laws which, when implemented in micro-macro approaches coupled with large-scale flow models, will provide more accurate estimations of the mechanical response of ice in the extreme conditions encountered in natural environments. These large-scale models will be able to accurately represent the texture evolution with strain and, hence, to take into account the mechanical anisotropy associated with the texture evolution with deformation in ice sheets [START_REF] Gillet-Chaulet | Flow-induced anisotropy in polar ice and related ice-sheet flow modelling[END_REF]. These new plasticity laws will also be able to tackle complex boundary conditions as the cyclic loading encountered in extraterrestrial bodies submitted to tidal forcing, as the saturnian satellite Enceladus [START_REF] Shoji | Ice rheology and tidal heating of Enceladus[END_REF].

Conclusions

The present study reveals for the first time the presence of a non-negligible (between 13% and 35%) proportion of dislocations with [c]-component Burgers vector within dislocation substructures in pure ice deformed in the laboratory at close to the melting temperature.

The characterization was made possible by the use of Weighted Burgers Vectors (WBV)

analyses that estimate the nature of geometrically necessary dislocations (GNDs) from "routine" EBSD measurements. This method is an alternative to classical techniques (X-ray diffraction, TEM) to identify the Burgers vector components of the GNDs, which has proven to be well adapted to the characterization of large, strongly deformed and recrystallized ice samples.

The fraction of dislocations with [c]-component Burgers vector in substructures is similar for various strain geometries and levels (compression or torsion creep, low to high strain). As [c]-component dislocations are energetically less favourable and possess higher Peierls barriers than < a > dislocations, they are expected in areas submitted to high local stresses.

Hence they should play an important role on the dislocation interactions during deformation (responsible for local hardening, internal stress field evolution), but also in the dynamic recrystallization processes (nucleation and grain boundary migration) that strongly impact microstructure and texture evolution in ice sheets. The present experimental evidence for activation of [c]-component dislocations in ice is a first, but essential step for perfecting the current crystal plasticity models and constraining the simulation of the role of these dislocations on the mechanical response of ice. To be able to represent this complexity in the chain of modeling tools that leads to the prediction of ice sheet and shelf flow is a step further toward an accurate prediction of their evolution in the frame of global climate changes. named after him), in general non-symmetric so having 9 independent components, and that this can be directly linked to the densities of GNDs and their line vectors. The idea is explained concisely in [START_REF] Sutton | Interfaces in crystalline materials[END_REF]. It provides in principle a powerful way of constraining possible GND types from lattice curvature, although there is not a unique way of deciding on dislocation types (lines and Burgers vectors) without further information or assumptions. Using EBSD data from 2D maps only 3 out of the 9 components of the tensor can be unambiguously determined without further assumptions, 1. In the differential form, local orientation gradients are used to calculate the WBV.

Acknowledgements

Errors are likely to be significant because of error-prone small misorientations, although Wheeler et al. (2009) show how they may be mitigated by filtering out the shortest WBVs. Adjacent measurement points with misorientations above a threshold value are omitted from gradient calculations, so as to exclude high angle boundaries which lack organised dislocation substructures. The magnitude and direction of the WBV can be displayed on maps in a variety of ways. Given that the shortest WBVs are the most error prone, the display may be chosen to show only those above a particular magnitude (cf. Fig 3a).

2. In the integral form, contour integration round the edge of a region on an EBSD map gives the net dislocation content of that region, though the spatial distribution of dislocations (domains of high or low density) within the region are not constrained.

The advantage is that errors are lower. This was asserted in (Wheeler et al., 2009) on the basis that numerical integration reduces the effects of noise, and has since been demonstrated using model EBSD maps for distorted lattices with added noise.

The method rejects any regions with high angle boundaries intersecting the border, using the threshold value mentioned above. The integral and differential methods are complementary and are built on the same mathematical foundation (they are linked via Stokes theorem).

  we used WBV analyses followingWheeler et al. (2009) (see Appendix A for a detailed description). The WBV represents the sum over different dislocation types of the product of [(density of intersections of dislocation lines within a selected area of the map) × (Burgers vector)]. Each dislocation line crossing the EBSD surface contributes to the WBV but the weight of this contribution depends of the angle between the dislocation line and the EBSD surface. It is one (zero) if the dislocation line is perpendicular to (within) the EBSD surface.

  to act as precursor of nucleation by strain induced boundary migration (SIBM) and bulging. The superposition of the WBV data (projection of the WBV on the sample plane and relative contribution of [c]-component dislocations, rW BV c ) to the trace of the SGBs (Fig. 1) highlights the complexity of the dislocation sub-structures and the variability of the contribution of [c]-component Burgers vector dislocations (from almost null to almost 1) in the different subgrain boundary segments.

Figure 1 :

 1 Figure 1: Serrated grain boundary observed in sample CI01. The ratio |W BVc|||WBV|| (see text) and the WBVs are plotted for the pixels where ||WBV|| is higher than 1.4 × 10 -3 µm -1 . The red arrows show the in-plane projections of the WBV direction (above a threshold of 1.4 × 10 -3 µm -1 , EBSD step size 5 µm).

Figure 2 :

 2 Figure 2: Weighted Burgers Vectors plotted over a zoomed area from the left side of figure 1, sample CI01.The colorscale gives the relative magnitude of [c]-component dislocations, rW BV c , and the red arrows show the in-plane projection of the WBV directions (above a threshold of 1.4 × 10 -3 µm -1 , EBSD step size of 5 µm). "Boundary plane" refers to "inferred" boundary plane, see text. Rectangular areas mark the domains selected for integral WBV calculations (table 2).

Figure 3 Figure 4 :

 34 Figure 3: a) Ratio between the W BV c component over the norm of the full WBV (rW BV c ) calculated at the pixel scale on sample TGI04 deformed in torsion up to γ = 1.94. As EBSD step size is 20 µm, a cut-off value of 3.5 × 10 -4 µm -1 was taken for ||WBV|| below which pixels are not considered in the calculation. b) Inverse Pole Figure color-coded EBSD image of the microstructure showing subgrain boundaries (in red) and grain boundaries (in black). c) Four small images to provide a focus on two illustrative cases, with the legend of respectively a) (left) and b) (right).

Figure 5 :

 5 Figure 5: Distribution of the WBV dominant component as a function of the pixel orientation characterised by its adapted Schmid factor (S = |σ.c| 2 -(c.σ.c) 2 , where σ is the stress tensor and c is the axis orientation), from the sample TGI03 deformed in torsion up to γ = 0.42. Top: c-axis pole figure and distribution of Schmid factors. Bottom: Ratio of pixels with ||WBV|| higher than 3.5 × 10 -4 µm -1 (EBSD step size of 20 µm). Each ratio is decomposed in 3 parts showing the dominant component of the WBV.

  The fact that we have observed for the first time a non-negligible contribution of [c]-component dislocations to the GNDs population in ice polycrystals deformed in the laboratory provides new constraints for modeling the deformation of ice. First, it gives a first order justification for the introduction of the activity of pyramidal slip that requires [c]-component dislocations into crystal plasticity laws. Indeed, Castelnau et al. (2008) and Suquet et al. (2012) both highlighted the necessity of a minimum amount of pyramidal slip to correctly simulate the behavior of ice polycrystals during transient creep by mean of full-field approaches. Secondly, the present observations open the possibility for a direct comparison between model predictions based on Dislocation Dynamics

Financial

  support by the French Agence Nationale de la Recherche is acknowledged (project DREAM, ANR-13-BS09-0001-01). This work benefited from support from institutes INSIS and INSU of CNRS. It has been supported by a grant from Labex OSUG@2020 (ANR10 LABEX56) and from INP-Grenoble and UJF in the frame of proposal called Grenoble Innovation Recherche AGIR (AGI13SMI15). Visiting exchanges for SP and MM were financed by ESF RPN MicroDICE (08RNP003) and by CCFS visiting research funds. MM benefited from a invited researcher fellowship from WSL, at SLF-Davos (2016-2017).

  indicates the Nth type of dislocation line, and for each type ρ is the density (m -2 ), b i the Burgers vector in crystal coordinates (m) and l γ the unit line vector in sample coordinates. As written the first index in α relates to the crystal reference frame and the second to sample reference frame and its units are m -1 .

but

  Wheeler et al. (2009) argued that even these three can provide valuable insights into possible dislocation types. Specifically the 3 components α i3 (where 3 indicates the sample coordinate direction perpendicular to the map) make up a vector related to the Burgers vectors of dislocations present. It is weighted with regard to the individual dislocation densities (through ρ) and the angles the dislocation lines make to the EBSD map (through l 3 ): hence Weighted Burgers Vector (WBV). For hexagonal phases such as ice the WBV can indicate the presence of vectors with a [c] component. Although the WBV does not record all the GNDs present, it cannot contain phantom directions. If it has a significant [c] component then at least some of the Burgers vectors of the GNDs must have a [c] component though this does not mean they have to be parallel to [c]. Wheeler et al. (2009) give two versions of the calculation.

Table 1 :

 1 Summary of the experimental conditions for the tests used in the study. Compression tests were performed under constant applied load, and torsion test under constant applied torque (the corresponding maximum shear stress is given here).

	• C	Stress	ε max γ max

Table 2 :

 2 Integral WBV projections over the four non-independent axes of the hexagonal crystal symmetry and the ratio rW BV c (see Materials and Methods), calculated for the areas of sample CI01 selected in figure2.

Institute of Engineering Univ. Grenoble Alpes Preprint submitted to Earth and Planetary Science Letters June

22, 2017

Appendix A. The Weighted Burgers Vector tool

Dislocations produce local distortions in crystal lattices. When dislocations of different signs are close together these distortions balance out and are not visible at the scale of microns. However when significant numbers of dislocations with the same signs are present, optically visible and (with EBSD) measurable variations of lattice orientation are a consequence the dislocations are then called geometrically necessary dislocations (GNDs). Crystalline materials generally have large elastic moduli meaning that lattice bending due to elastic stress is likely to be small; significant curvature generally relates to the presence of GNDs. [START_REF] Nye | Some geometrical relations in dislocated crystals[END_REF] recognized that the lattice curvature can be described by a second rank tensor (now In this contribution we discuss subgrain boundaries (SGBs). As happens in many materials, GNDs have moved by recovery into discrete structures. As these are two dimensional features, with zero volume, then strictly the dislocation density is infinite. However the integral method still gives a rigorous measure of the dislocation content within a region, if that region includes a subgrain boundary: [START_REF] Sutton | Interfaces in crystalline materials[END_REF] show how closely the analysis of SGBs relates to the analysis of smoothly curved lattices. Hence the direction of the integral WBV still carries useful information related to the GNDs in SGBs. We show colour coded maps of the magnitude of the differential form of the WBV. When this is calculated, numerical differentiation is used. Suppose we have two measurement points with 2.5 • difference in orientation separated by a 5 µm step size, then the calculated orientation gradient will be 0.5 • / µm. This may in reality be a smoothly curved lattice, or relate to a sharp 2.5 • SGB passing between the two measurement points the method cannot distinguish such possibilities. If it is an SGB then a smaller step size of 2.5 µm would give rise to an apparent gradient of 1 • / µm. Consequently around SGBs the magnitude of the WBV depends on step size (and hence should be interpreted with caution) but the direction can still be used to constrain GND types.

The disadvantages of the WBV approach are: it is less precise than calculations using high (angular) resolution EBSD [START_REF] Wallis | Geometrically necessary dislocation densities in olivine obtained using high-angular resolution electron backscatter diffraction[END_REF], it is biased towards dislocation lines intersecting the EBSD map at a high angle, and it does not give a decomposition of the GND population into different dislocation types. The latter can be attempted by making particular assumptions about the dislocation types present and then making a calculation assuming total dislocation energy is minimised. As argued in Wheeler et al. (2009), though, minimising energy without taking into account elastic interactions between dislocations (which will mean that line energies are not simply additive) may not be an appropriate procedure.

The advantages of the WBV are: it can be calculated from routinely collected EBSD data, in a way free from assumptions except that the elastic strains be small. The integral form reduces the propagation of errors inherent in Kikuchi pattern indexing, and can be used to analyse both smoothly curved lattices and SGBs, without any assumptions about twist or tilt nature. As this contribution shows the WBV approach is sufficient to test the hypothesis that dislocations with [c] component Burgers vectors in ice form a significant part of the dislocation substructures.