Christophe Vuillot

IS ERROR DETECTION HELPFUL ON IBM 5Q CHIPS ?

Keywords: quantum, fault-tolerance, error detection, experiment

This paper reports on experiments realized on several IBM 5Q chips which show evidence for the advantage of using error detection and fault-tolerant design of quantum circuits. We show an average improvement of the task of sampling from states that can be faulttolerantly prepared in the [[4, 2, 2]] code, when using a fault-tolerant technique well suited to the layout of the chip. By showing that fault-tolerant quantum computation is already within our reach, the author hopes to encourage this approach.

Introduction

Quantum systems in laboratories around the world are reaching unprecedented level of control and precision for a variety of devices aiming at implementing reliable qubits. Yet, due to the very nature of those devices, errors are still notably present. In order to be able to execute long quantum algorithms, improvements should be made using quantum error correction and fault-tolerant schemes. So-called threshold theorems [START_REF] Aharonov | Fault-tolerant quantum computation with constant error[END_REF][START_REF] Knill | Resilient quantum computation: error models and thresholds[END_REF] prove that there exist error rates below which this approach is guaranteed to improve the device performance. However, between those theorems and real experiments there remains a fog of technical details which clouds the actual practicality of error correcting codes.

Already some experiments which demonstrate the usefulness of quantum error correcting codes to protect a quantum memory have been done, e.g. [START_REF] Knill | Benchmarking quantum computers: The five-qubit error correcting code[END_REF][START_REF] Nigg | Quantum computations on a topologically encoded qubit[END_REF][START_REF] Córcoles | Demonstration of a quantum error detection code using a square lattice of four superconducting qubits[END_REF][START_REF] Linke | Fault-tolerant quantum error detection[END_REF][START_REF] Takita | Experimental demonstration of fault-tolerant state preparation with superconducting qubits[END_REF], but a demonstration of protected computation is still missing.

Inspired by a recent proposal by D. Gottesman [START_REF] Gottesman | Quantum fault tolerance in small experiments[END_REF], we use an IBM 5Q chip to show that error detection can improve some simple sampling tasks, thus turning a tiny portion of fog into blue sky. Closely related to this work, [START_REF] Linke | Fault-tolerant quantum error detection[END_REF] and [START_REF] Takita | Experimental demonstration of fault-tolerant state preparation with superconducting qubits[END_REF] present experiments based on the same error detecting code, on trapped-ion qubits and on a similar superconducting chip, respectively. Both those works focus on the preparation of few states and only one of the two encoded qubits is handled fault-tolerantly. In this work an extensive set of fault-tolerant circuits is studied and both logical qubits are protected. Moreover [START_REF] Linke | Fault-tolerant quantum error detection[END_REF] and [START_REF] Takita | Experimental demonstration of fault-tolerant state preparation with superconducting qubits[END_REF] introduce artificial Pauli errors to study the robustness of their preparation whereas this work takes a higher level a c.vuillot@tudelft.nl approach, treating the experimental set-up as a black box, probing only the intrinsic errors in the system. These considerations make the present work closer to the spirit of [START_REF] Gottesman | Quantum fault tolerance in small experiments[END_REF]. This work is, to the author's knowledge, the first experimental demonstration of error detection and fault-tolerance improving a quantum computation.

The paper is organized as follows. In section 2 we present the principle of the approach and its specialization to the IBM 5Q chip. In section 3 the experimental results are shown and analyzed.

Demonstrating fault-tolerance

The idea behind fault-tolerance is to devise error correcting codes and the corresponding processes of encoding, correcting, computing or measuring with encoded information such that more errors are corrected than introduced [START_REF] Shor | Fault-tolerant quantum computation[END_REF]. The difficulty in devising such processes is that they are built out of components that are all faulty, so adding some might do more harm than it can help. Devising, proving and simulating fault-tolerant schemes has been pursued for the last 20 years, e.g. [START_REF] Aharonov | Fault-tolerant quantum computation with constant error[END_REF][START_REF] Shor | Fault-tolerant quantum computation[END_REF][START_REF] Aliferis | Quantum accuracy threshold for concatenated distance-3 codes[END_REF][START_REF] Aliferis | Accuracy threshold for postselected quantum computation[END_REF][START_REF] Cross | A comparative code study for quantum fault tolerance[END_REF][START_REF] Paetznick | Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit golay code[END_REF][START_REF] Tomita | Low-distance surface codes under realistic quantum noise[END_REF][START_REF] Chamberland | Thresholds for universal concatenated quantum codes[END_REF], but improving and finding better schemes is still important and a subject of ongoing research, e.g. [START_REF] Chao | Quantum error correction with only two extra qubits[END_REF][START_REF] Chao | Fault-tolerant quantum computation with few qubits[END_REF][START_REF] Takagi | Error rates and resource overheads of encoded three-qubit gates[END_REF][START_REF] Chamberland | Overhead analysis of universal concatenated quantum codes[END_REF][START_REF] Chamberland | Flag fault-tolerant error correction with arbitrary distance codes[END_REF].

Taking one of these schemes and experimentally demonstrating its fault-tolerance, i.e finding an improvement of performance going from bare to encoded implementation, provides the ultimate validation of the scheme.

General approach

A description of a general approach to demonstrate fault-tolerant quantum computation is given in [START_REF] Gottesman | Quantum fault tolerance in small experiments[END_REF]; we just briefly recall it here. The idea is to choose a quantum error correcting code C, admitting fault-tolerant circuits for the preparation of some logical states, denoted as |s 1 , . . . , |s m ∈ S C FT , as well as for some logical quantum gates, denoted as U 1 , . . . , U n ∈ G C FT . Using these as building blocks one can then randomly draw a state preparation from S C FT , then draw a sequence of gates from G C

FT to obtain an encoded fault-tolerant circuit. More precisely, following the formalism of rectangles and extended rectangle introduced in [START_REF] Aliferis | Quantum accuracy threshold for concatenated distance-3 codes[END_REF], one interleaves the logical units (gates in G C FT , or preparation of states in S C FT ) with a fault-tolerant circuit for error correction. Rectangles designate circuits comprising one logical unit preceded by a round of error correction, extended rectangles designate circuits comprising one logical unit preceded and followed by a round of error correction. The precise conditions to satisfy in order to be fault-tolerant for distance 3 codes are stated in [START_REF] Aliferis | Quantum accuracy threshold for concatenated distance-3 codes[END_REF]. This ensures that the whole computation is error free if there is at most one fault per extended rectangle.

The error model considered in this paper is that the failure of any component can introduce any Pauli error on the qubits acted on by the component. In our case, we cannot repeat faulttolerant detection of errors, so our circuits will only tolerate a single fault in total.

The encoded circuits then have to be compared to a bare implementation on the physical qubits. Sampling from the final state produced by the circuit in the computational basis and comparing the probability distribution obtained with the expected one gives a simple comparison metric. Note that this supposes that the circuits are small enough or simple enough so that one can classically simulate sampling from the final state efficiently. A successful demonstration of fault-tolerant quantum computation happens if the encoded circuits show better performance than the bare circuits. To be the most convincing, one needs to use the best of the physical qubits as well as the most efficient implementation for the bare circuits.

2.2

The IBM 5Q chip and [[4,2,2]] (a) Layout In 2016 IBM released a quantum chip with fixed-frequency superconducting transmon qubits, named IBM 5Q. They provide worldwide cloud access to the chip under an initiative called the "IBM Quantum Experience" [START_REF]IBM Quantum Experience[END_REF]. The current iteration on which the experiments were done is nicknamed Raven. The appendix also presents and compares preliminary results obtained with a previous iteration, Sparrow. The chip has five qubits, natively named q 0 to q 4 . It features single qubit Clifford gates, the T gate (diag(1, e iπ/4 )) as well as some two-qubits CNOTs with a certain layout represented in Fig. 1a. This many qubits is the right number to use for a demonstration using the [[4, 2, 2]] code as discussed in [START_REF] Gottesman | Quantum fault tolerance in small experiments[END_REF].

q 2 4 q 1 1 q 3 2 q 4 3 q 0 A (b) [[4, 2, 2]] Code 1 2 3 4 S X , S Z Z 2 , X 1 Z 1 , X 2
The [[4, 2, 2]] code encodes two qubits into four physical qubits. Its code space is stabilized by the all-X and all-Z Pauli operators (S X = X ⊗ X ⊗ X ⊗ X, S Z = Z ⊗ Z ⊗ Z ⊗ Z), together with the logical Pauli operators, they are represented in Fig. 1b. The logical code states are

|00 L = 1 √ 2 (|0000 + |1111 ) (1)
|01 L = 1 √ 2 (|1100 + |0011 ) (2)
|10 L = 1 √ 2 (|1010 + |0101 ) (3)
|11 L = 1 √ 2 (|1001 + |0110 ) . (4)
The code also admits transversal implementations of two Clifford gates, namely H ⊗H •SWAP, where H is the Hadamard gate, see Fig. 2a, and the controlled phase or CZ gate, see Fig. 2b. Moreover, if the five qubits have the right connectivity then there are fault-tolerant circuits preparing the logical states |00 L , |0+ L and |00 L + |11 L . For the cat state |00 L , see Equation ( 1), there exists fault-tolerant preparations including one ancillary qubit to verify the preparation. One post-selects on a successful preparation. The logical states |0+ L and |00 L + |11 L are both two Bell pairs but with different pairings. They both also have faulttolerant preparation circuits. In [START_REF] Gottesman | Quantum fault tolerance in small experiments[END_REF], some circuits are proposed, they can be adapted to the layout of the chip. We tested another preparation for |00 L inspired by [START_REF] Chao | Quantum error correction with only two extra qubits[END_REF], which is substantially shorter so more relevant to this layout.

(a) (a) FTv1 In both circuits if the ancillary qubit q 0 is measured as 1, we reject the preparation.

q 1 H H × ≡ q 2 H H × q 3 H q 4 H (b) q 1 S • ≡ q 2 S H H q 3 S q 4 S
q → |0 ✌ ✌ ✌ q → |0 q → |0 H • • H H • • q → |0 H • H q → |0 (b) FTv2 |0 ✌ ✌ ✌ |0 • |0 H • × • |0 H • H • |0 ×
The flagging technique in [START_REF] Chao | Quantum error correction with only two extra qubits[END_REF] for the preparation of |00 L can be implemented with the given CNOT connectivity, see Fig. 3a. We call this the fault-tolerant version one (FTv1). The technique based on a circular layout proposed by [START_REF] Gottesman | Quantum fault tolerance in small experiments[END_REF] cannot be implemented directly due to the connectivity. A clever introduction of one SWAP gate permits to implement it still fault-tolerantly at the cost of 8 CNOT gates in total, see Fig. 3b. We call this the fault-tolerant version two (FTv2). We present below, the results for FTv1, which is more appropriate for the layout. For both those circuits one can check that every possible single fault leads to a detectable error, or an error that stabilizes the prepared state. We also tried a non-fault-tolerant one that has the advantage of being short, involving only 3 CNOTs, see Figure 4. We call this the non-fault-tolerant version (NFT). For the preparation of |0+ L , we want to create a Bell pair between q 1 and q 3 and between q 2 and q 4 . There is a missing connection between q 1 and q 3 , but we can do a similar SWAP trick, see the resulting circuit in Fig. 6a. This circuit is not fault-tolerant as an undetected logical

Z 1 ⊗ Z 2 = Z ⊗ 1 ⊗ 1 ⊗ Z can occur if the SWAP
gate fails. This is the only possible harmful logical error for this circuit. We'll see below that we only use this circuit in cases were the possible undetected Z 1 ⊗ Z 2 error doesn't change the outcome of the sampling test. The preparation for |00 L + |11 L can itself be straightforwardly implemented fault-tolerantly, see Fig. 6b. To summarize, our sets of initial states and gates are

S FT = |00 , |0+ , |00 + |11 √ 2 , G FT = {X 1 , X 2 , Z 1 , Z 2 , H ⊗ H • SWAP, CZ} .
The [[4, 2, 2]] code is only an error detecting code, that is, it can detect one Pauli error but cannot correct it. This means that in place of error correction we have to rely on postselection to remove errors. In other words, we throw away runs where we detect that an error occurred, either from the ancilla measurement checking state preparation, or from the final measurement which indicates, when the parity of the outcomes is odd, that S Z has value -1.

q 0 → |0 q 1 → |0 q 2 → |0 • q 3 → |0 H • • q 4 → |0
As mentioned before, we cannot interleave rounds of error detection between state preparation and the gates. That means that the final circuit can only be tolerant to a single fault during the whole computation, except for the specific undetectable Z 1 ⊗ Z 2 failure mentioned above and when we use the non-fault-tolerant version to prepare |00 L . (a)

× = • H • H • × H H
q 1 → |0 H × q 2 → |0 × • q 3 → |0 H • q 4 → |0 (b) |0 |0 H • |0 H • |0

Comments on the tested circuits

Since we work with such a small system, we can exhaustively find and try all the logically equivalent circuits and optimize the bare version for each one. Essentially, we are making it most likely for the bare version of the task to prevail. From the set of states S FT , and the set of gates G FT , one can obtain 20 different stabilizer states. All the states with their most efficient bare preparation circuit, using the native set of gates provided by IBM, are listed in Figure 7. A brute force approach was used to find them. Note that adding more gates in the sequence would test states that are already tested with shorter circuits. Therefore those would certainly give worse results as we cannot interleave error detections between gates. Hence we kept only these 20 different preparations. This still extensively probes the computational capabilities of the [[4, 2, 2]] code.

Initial state

Unitary Final state # Instructions

|00 1 ⊗ 1 |00 |0+ 1 ⊗ 1 |00 +|01 √ 2 |00 1 ⊗ X |01 |00 X ⊗ 1 |10 |00 +|11 √ 2 1 ⊗ 1 |00 +|11 √ 2 |0+ 1 ⊗ Z |00 -|01 √ 2 |00 H ⊗ H • SWAP |00 +|01 +|10 +|11 2 |0+ X ⊗ 1 |10 +|11 √ 2 |00 X ⊗ X |11 |00 +|11 √ 2 1 ⊗ Z |00 -|11 √ 2 |00 +|11 √ 2 X ⊗ 1 |10 +|01 √ 2 |00 1 ⊗ Z • H ⊗ H • SWAP |00 -|01 +|10 -|11 2 |00 Z ⊗ 1 • H ⊗ H • SWAP |00 +|01 -|10 -|11 2 |0+ X ⊗ Z |10 -|11 √ 2 |00 +|11 √ 2 1 ⊗ ZX |10 -|01 √ 2 |00 Z ⊗ Z • H ⊗ H • SWAP |00 -|01 -|10 +|11 2 |00 CZ • H ⊗ H • SWAP |00 +|01 +|10 -|11 2 10 |00 CZ • 1 ⊗ Z • H ⊗ H • SWAP |00 -|01 +|10 +|11 2 11 |00 CZ • Z ⊗ 1 • H ⊗ H • SWAP |00 +|01 -|10 +|11 2 11 |00 1 ⊗ X • CZ • H ⊗ H • SWAP • X ⊗ 1 |00 -|01 -|10 -|11 2
12 Fig. 7. The list of initial states and unitary circuits and the number of QASM instructions in the bare circuit (including final measurements). The final states are written exclusively in the computational basis since that is how they are measured.

Experimental results

The code and data can be found on GitHub [START_REF]Code and data of the experiments[END_REF]: it uses the Python SDK that can be found at [START_REF]Python sdk[END_REF].

Parameters and runs

Using the IBM chip one can run circuits in batches. For each individual run, 8192 shots are done right one after another in a short time, where each shot outputs 5 bits as measurement outcomes. We consider that the chip is exactly in the same conditions during each run.

We consider all the runs to be independent of one another but each to be done in different conditions, so sampling from different output distributions.

For each run, IBM also provides calibration data describing the state of the chip such as: gate error rates, readout error rates, T 1 , T 2 and fridge temperature. We give the average values that we observed for our runs in the appendix.

Performance metric

For each circuit run we want to compare the observed outcome distribution with the ideal one. The ideal distribution is 8192 independent samples from a distribution with four possible outcomes occurring with probabilities p 00 , p 01 , p 10 and p 11 . The values for p ij can be read from Figure 7. Since we assume that the conditions stay identical during one run and that the 8192 shots are independent, we observe independent samples from a four-outcome distribution with some different probabilities p00 , p01 , p10 and p11 .

We then use the statistical distance as a metric:

D = 1 2 |p 00 -p00 | + |p 01 -p01 | + |p 10 -p10 | + |p 11 -p11 | .
This quantity is estimated for each run by where n ij is the number of observation of outcome ij after post-selection and n valid is the number of shots kept after post-selection. This estimator for D is slightly biased except for the case where only one of the p ij is non-zero (because in this case it becomes linear). We use this estimator to keep the analysis simple.

D =
Each of the runs has some different pij and we have no information about how the pij s vary. Therefore we will assume that there are fluctuations around the mean of D following some unknown normal distribution and use this model to compute confidence intervals [START_REF] Walpole | Probability and Statistics for Engineers and Scientists[END_REF]. This means that the final data points and their confidence intervals don't exactly reflect knowledge of D but only of D which we believe is still a valid quantity to characterize the performance of the circuits.

Comparisons

We first need to decide on what pair of bare qubits is the best to be compared to the encoded qubits. It is not immediately clear how to choose this given only the calibration numbers. Thus we tried out the six different connected pairs and we found the performance shown in Figure 8. There is no clear "best pair" but one can see that the pairs [2 -0] or [3 -2] seem to be slightly better. Since we don't have a systematic method for predicting the best pair given the calibration data and the circuit, we will look at average performance for each pair over all the circuits. When averaging, we find the pair [2 -0] to be the best, see Table 1. The second observation that we can make based on Figure 8 is that the number of instructions does not explain the performance. It seems that the type of state sampled from is more important. Roughly it is easier to sample from equal superposition of the four computational basis states, than from equal superposition of two, than from just one. This can be understood with the fact that with more states in equal superposition there are less Pauli errors or readout errors that can affect the outcome distribution.

#instructions |0 0 > |0 + > X 2 |0 0 > X 1 |0 0 > |0 0 > + |1 1 > Z 2 |0 + > H H S |0 0 > X 1 |0 + > X 2 -X 1 |0 0 > Z 2 |0 0 > + |1 1 > X 1 |0 0 > + |1 1 > Z 2 -H H S |0 0 > Z 1 -H H S |0 0 > Z 2 -X 1 |0 + > X 2 -Z 2 |0 0 > + |1 1 > Z 2 -Z 1 -H H S |0 0 > C Z -H H S |0 0 > C Z -Z 2 -H H S |0 0 > C Z -Z 1 -H H S |0 0 > X 2 -C Z -H H S -X 1 |0 0 > 0
We then go on, comparing the encoded versions of the circuits to the elected best pair in Figure 9. The encoded circuits using the preparations |0+ L and |00 L + |11 L are short and fault-tolerant and indeed show better performance than the bare ones. For the different preparations for |00 L , the fault-tolerant version FTv1 is, except in a few cases, better than the non-fault-tolerant one despite being substantially longer. This shows that fault-tolerant design of circuits can be useful. Although they both compare unfavourably to the bare version, except for 4 circuits for FTv1 and only 1 circuit for NFT.

We cannot make an absolute statement as for some circuits the bare version is always better. We also don't want to cherry pick the best version for each circuit since we don't have a systematic way of predicting the best version. Average performance when fixing a

|0 0 > |0 + > X 2 |0 0 > X 1 |0 0 > |0 0 > + |1 1 > Z 2 |0 + > H H S |0 0 > X 1 |0 + > X 2 -X 1 |0 0 > Z 2 |0 0 > + |1 1 > X 1 |0 0 > + |1 1 > Z 2 -H H S |0 0 > Z 1 -H H S |0 0 > Z 2 -X 1 |0 + > X 2 -Z 2 |0 0 > + |1 1 > Z 2 -Z 1 -H H S |0 0 > C Z -H H S |0 0 > C Z -Z 2 -H H S |0 0 > C Z -Z 1 -H H S |0 0 > X 2 -C Z -H H S -X 1 |0 0 > -0.1 -5 • 10 -2 0 5 • 10 -2 0.1
Differences of performance (Encoded -Bare)

|00 FTv1 |00 NFT |0+ |00 + |11
preparation for |00 L are shown in Table 1. One can see that FTv1, with 4.51 × 10 -2 , is better than the best bare implementation with 5.5 × 10 -2 , whereas NFT (6.05 × 10 -2 ) is worse.

Conclusion

In conclusion, we have shown that already on the IBM 5Q chip one can improve some quantum computation task, namely sampling from a class of states, by using error detection and fault-tolerant design of circuits. This improvement is only on average over 20 different states that the [[4, 2, 2]] code can fault-tolerantly prepare. We also can see that shorter but non fault-tolerant circuits can be bested by longer but fault-tolerant circuits, showing the usefulness of these. As better and better hardware is developed, with more physical qubits, more connectivity and smaller error rates, demonstrations of fault-tolerance will become easier to produce and become more convincing. The set of gates shown to be fault-tolerant in this paper is very restricted. For the [[4, 2, 2]] code a few more qubits and connections would be needed to realize fully fault-tolerant circuits with error detection in between logical units. Being able to demonstrate the fault-tolerance of larger gate sets, for example the whole Clifford group for several logical qubits would be an important milestone towards harnessing universal Implementation Avg.Perf.(×10

q 0 → |0 H • • H ✌ ✌ ✌ q 1 → |0 × H H q 2 → |0 × H H q 3 → |0 H • • q 4 → |0 • Fig. A.2.
The circuit previously implemented on the IBM 5Q Sparrow chip preparing the logical state |00 L . The SWAP gate is implemented via the circuit in Figure 5. If the SWAP gate fails, it can introduce Pauli X errors on q 1 and q 2 , which would not be detected and which constitute a logical X 1 ⊗ X 2 error.

(a) Except for the four circuits involving a two-qubit gate, Sparrow was showing better performance. This is probably due to improvement of the two-qubit gates but shorter T 1 and T 2 times on Raven, see Table A

q 1 |0 × q 2 |0 × q 3 |0 H • q 4 |0 H • (b) |0 H • |0 |0 H • |0

Fig. 1 .

 1 Fig. 1. (a) The connectivity between the qubits in the IBM 5Q Raven chip. Arrows indicate CNOT capabilities where the arrow points towards the target of the CNOT. Red labels indicate our choice of qubit numbering. A is an ancillary qubit used in verification of state preparation. (b) The usual representation layout for the [[4, 2, 2]] code. The numbered circles represent the qubits and the rectangular boxes represent the support of different Pauli operators: logical Pauli operators Z 1 , X 2 are highlighted in blue, logical Pauli operators Z 2 , X 1 in green and stabilizers S X , S Z in black.

Fig. 2 .

 2 Fig. 2. (a) Bare and encoded versions of H ⊗H •SWAP, where H is the Hadamard gate. In practice the SWAP gate is not done physically on the bare version but just in software, by renumbering the two qubits. (b) Bare and encoded versions of the CZ gate, where S = diag(1, i).

Fig. 3 .

 3 Fig. 3. The two different implementations for preparing the state |00 L : (a) Based on a flagging technique with 5 CNOTs, (b) based on the circular connectivity circuit with 8 CNOTs.In both circuits if the ancillary qubit q 0 is measured as 1, we reject the preparation.

Fig. 4 .

 4 Fig. 4. A short but non-fault-tolerant circuit to prepare |00 L .

Fig. 5 .

 5 Fig. 5. Implementation of the SWAP gate.

Fig. 6 .

 6 Fig. 6. (a) Preparation circuits of the logical state |0+ L , with 5 CNOTs. (b) Preparation circuits of the logical Bell state |00 L + |11 L / √ 2 with 2 CNOTs.

Fig. 8 .

 8 Fig. 8. Comparison of the different pairs of qubits to implement the bare circuits together with the number of instructions for each circuit.

Fig. 9 .

 9 Fig. 9. Comparing encoded performance with bare qubit pair [2-0]. The performance is defined as the statistical distance to the ideal outcome distribution. The figure shows the difference between encoded and bare performance. The error bars show confidence intervals at 99%.

Fig. A. 3 .

 3 Fig. A.3. (a) Preparation circuits of the logical state |0+ L , with 5 CNOTs. If the SWAP gate fails, it can introduce Pauli X errors on q 1 and q 2 , which would not be detected and which constitute a logical X 1 ⊗ X 2 error. (b) Preparation circuits of the logical Bell state |00 L + |11 L / √ 2 with 2 CNOTs.

Fig. A. 4 .

 4 Fig. A.4. Comparison of the performances of the bare circuits between the Raven and Sparrow chips.Except for the four circuits involving a two-qubit gate, Sparrow was showing better performance. This is probably due to improvement of the two-qubit gates but shorter T 1 and T 2 times on Raven, see TableA.2 and Table A.4.

Fig. A. 5 .

 5 Fig. A.5. Comparing encoded versions on Raven and Sparrow. The |00 L and |0+ L preparations were suffering from possible X 1 X 2 logical errors.

Table 1 .

 1 Average performance for the different bare and encoded possible implementations with the post-selection ratio (ratio of data kept). The encoded implementations only differ in how they prepare the state |00 L .Table A.3. Average single-qubit gate and readout error rates for the runs together with their standard deviation, all in percent.

			-2 ) Post-selection ratio
	Bare[1-0]	6.72	1.00
	Bare[2-0]	5.50	1.00
	Bare[2-1]	6.49	1.00
	Bare[2-4]	6.98	1.00
	Bare[3-2]	6.27	1.00
	Bare[3-4]	7.73	1.00
	FTv1	4.51	0.65
	NFT	6.05	0.71
	quantum computation.

Table

A

.

[START_REF] Nigg | Quantum computations on a topologically encoded qubit[END_REF]

. Average CNOT gate error rates and their standard deviation for the runs, all in percent.

  .2 and Table A.4.

Acknowledgements

I would like to thank the IBM Quantum Experience team for providing extensive access to the chip and support on how to interface with it. I also would like to thank Barbara Terhal for valuable comments and feedback. CV acknowledge support through the EU via the ERC GRANT EQEC and support by the Excellence Initiative of DFG. The views expressed are those of the author and do not reflect the official policy or position of IBM or the IBM Quantum Experience team.

Appendix A

This appendix presents the calibration data for the experiments, averaged over all the different runs as well as the observed standard deviation. It also shows previous runs on a previous iteration of the chip, called Sparrow, which was showing better performance but has been taken down by IBM. The current iteration is named Raven: the more extensive runs presented in this paper were done on this chip. Note that for the runs on Sparrow, there were only 4 different calibrations and 36 runs for each circuit. For Raven we have more than 100 runs for each circuit and almost a new calibration per run. The two chips are similar, the gate and readout error rates have been slightly improved with Raven but the fridge temperature as well as T 1 and T 2 times became a bit worse. The layout differs only in the orientation of the CNOTs, see


Martina Caramenti
email: martina.caramenti@ibfm.cnr.it

Claudio L Lafortuna
email: claudio.lafortuna@cnr.it

Elena Mugellini
email: elena.mugellini@hes-so.ch

Omar Abou Khaled
email: omar.aboukhaled@hes-so.ch

Jean-Pierre Bresciani
email: jean-pierre.bresciani@unifr.ch

Amandine Dubois
email: amandine.dubois@univ-lorraine.fr

No Evidence That Frontal Optical Flow Affects Perceived Locomotor Speed and Locomotor Biomechanics When Running on a Treadmill

Keywords: virtual reality, visual speed perception, treadmill running, self-motion perception, optical flow, locomotion, biomechanics

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Human locomotion is controlled through a complex sensorial integration of different types of information, in particular about the relative motion between the body and the surrounding environment [START_REF] Sun | Multisensory integration in the estimation of relative path length[END_REF]. Visual, vestibular, proprioceptive, and auditory information is integrated by the central nervous system, allowing the generation of the perception of motion [START_REF] Mergner | Interaction of vestibular, somatosensory and visual signals for postural control and motion perception under terrestrial and microgravity conditions-A conceptual model[END_REF]. Though walking is influenced by different factors [START_REF] Dietz | Proprioception and locomotor disorders[END_REF], the main contribution derives from visual, vestibular and proprioceptive signals. Using static cues, object-motion cues and optical flow cues, the visual system allows us to discriminate self-motion from movement of objects of the environment [START_REF] Sun | Multisensory integration in speed estimation during self-motion[END_REF]. Vestibular cues provide motion data about changes of velocity or direction-based data about accelerations [START_REF] Angelaki | Vestibular system: The many facets of a multimodal sense[END_REF]. Lastly, the proprioceptive system uses joint-and muscle-related information to provide feedback about the status of the whole musculoskeletal system, which greatly contributes to the perception and control of motion during active movements [START_REF] Dietz | Proprioception and locomotor disorders[END_REF]. Speed perception, goal-directed movements, navigation, and collision avoidance all rely on the congruency and integration of the signals provided by these different sensory systems.

Visual information plays an important role in motion perception. In particular, optical flow information is used to estimate direction [START_REF] Warren | Direction of self-motion is perceived from optical flow[END_REF], egocentric speed [START_REF] Larish | Sources of optical information useful for perception of speed of rectilinear selfmotion[END_REF], and time-to-collision [START_REF] Lee | A theory of visual control of braking based on information about time-to-collision[END_REF], and it heavily contributes to heading perception [START_REF] Bruggeman | Optic flow drives human visuo-locomotor adaptation[END_REF][START_REF] Rushton | Guidance of locomotion on foot uses perceived target location rather than optic flow[END_REF][START_REF] Warren | Optic flow is used to control human walking[END_REF]. The absolute travelled distance can also be calculated by integrating this information [START_REF] Bremmer | The use of optical velocities for distance discrimination and reproduction during visually simulated self motion[END_REF][START_REF] Riecke | Visual homing is possible without landmarks: A path integration study in virtual reality[END_REF]. However, when visual cues are in conflict with other sensory information about self-motion, visual information not always constitutes the "dominant" source of information [START_REF] Sun | Multisensory integration in speed estimation during self-motion[END_REF][START_REF] Harris | Visual and non-visual cues in the perception of linear self motion[END_REF]. In fact, when walking at constant speed, people tend to weight the information deriving from proprioceptive cues higher than visual cues [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF][START_REF] Varraine | Interaction between different sensory cues in the control of human gait[END_REF][START_REF] Pailhous | Unintentional modulations of human gait by optical flow[END_REF]. Nevertheless, visual input seems to have a great impact on self-motion perception while walking, and studies have shown how the manipulation of the optical flow affects visuomotor recalibration [START_REF] Warren | Optic flow is used to control human walking[END_REF][START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF][START_REF] Pailhous | Unintentional modulations of human gait by optical flow[END_REF][START_REF] Durgin | The perception of linear self-motion[END_REF][START_REF] Mohler | Visual motion influences locomotion in a treadmill virtual environment[END_REF]. In particular, it was found that if optical flow is manipulated to create sensory conflicts when walking, the locomotor speed decreases as the optical flow speed increases [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF][START_REF] Konczak | Effects of optic flow on the kinematics of human gait: A comparison of young and older adults[END_REF], suggesting a response to the relative weightings assigned to different sensory cues, in this case proprioceptive and visual cues [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF]. Some studies have also shown that the manipulation of optical flow can significantly influence locomotor patterns [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF][START_REF] Pailhous | Unintentional modulations of human gait by optical flow[END_REF][START_REF] Baumberger | Walking in an environment of moving ground texture[END_REF][START_REF] Zijlstra | Voluntary and involuntary adaptation of walking to temporal and spatial constraints[END_REF].

All above-mentioned studies were performed with walking participants and, to our knowledge, the influence of optical flow on self-motion perception and on locomotor patterns has never been investigated with running individuals. Yet, locomotor patterns are modulated differently in walking and running [START_REF] Brandt | Vestibulopathic gait: you're better off running than walking[END_REF][START_REF] Jahn | Visually induced gait deviations during different locomotion speeds[END_REF][START_REF] Jahn | Brain activation patterns during imagined stance and locomotion in functional magnetic resonance imaging[END_REF], with automated spinal programs allowing path integration in running, while walking is more dependent on visual control [START_REF] Jahn | Visually induced gait deviations during different locomotion speeds[END_REF]. For this reason, one might wonder whether the results obtained with walking individuals can be generalized to running individuals.

Here we investigated how the presence and manipulation of a virtual optical flow influences the control of locomotion while running on a treadmill. In Experiment 1, we tested how the presence and manipulation of a virtual optical flow influences self-motion speed perception. Specifically, participants were instructed to retrieve a previously presented reference running speed while being exposed to different optical flow conditions. In Experiment 2, we investigated how the presence and manipulation of an artificial optical flow influences gait parameters while treadmill running at a selfselected constant speed with different optical flow conditions.

Experiment 1

Introduction

In this experiment, we tested if the perception of self-motion speed is influenced by the presence and the manipulation of an optical flow. For walking, studies have shown that movement influences the perception of the optical flow speed in virtual environments [START_REF] Durgin | Perception of visual speed while moving[END_REF], and that the rate of optical flow has a modulating effect on walking speed, with an inverse linear relationship between optical flow and walking velocity [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF]. Optical flow has also been shown to affect stride length, cadence, and velocity [START_REF] Pailhous | Unintentional modulations of human gait by optical flow[END_REF]. All these results suggest an internal calibration between locomotion and visual perception. Here we investigated if participants were able to retrieve a previously experienced running speed in different optical flow conditions while running on a treadmill.

Methods

Participants

12 healthy participants (4 female, 8 male) with a mean age of 25.0 (±1.2 SD) participated in this study. They were unaware of the purpose of the study, were moderately trained to trained, had normal or corrected-to-normal vision, and none had a history of cardiovascular disease. All participants gave their informed and written consent prior to the inclusion in the study. This study was performed in accordance with the ethical standards specified by the 1964 Declaration of Helsinki and approved by the Ethics Committee of the University of Fribourg.

Experimental Setup

Participants ran on a HP Cosmos Mercury treadmill with a running surface of 150 × 50 cm. The treadmill was positioned in front of a 4.30 × 2.70 m screen at a distance of 1.7 m, leading to an effective field of view of 90°. To simulate optic flow, a virtual environment was projected onto the screen using a Barco F50 WUXGA projector with a 1920 × 1200 pixels resolution. The VR scene for which judgements were made and that was subsequently manipulated was created using Unity and depicted a neutral open-air hallway presented at constant-velocity motion (Figure 1). Rich optical flow information was provided by a granular texture on the floor and by the random pattern on the walls, without giving any landmarks or usable spatial information. The room was darkened during the experiment, with the display screen was the only source of light. While running, participants wore a soundproof headset (Hearing Protection type Pamir-Swiss Army).

Procedure

Prior to the running task, the participants filled out a custom-made questionnaire about their sports activity.

Before starting the actual test, participants familiarized themselves with treadmill running and experimental trials were initiated when the participant felt comfortable. Every single trial would start with the participant running at constant speed. Participants were instructed to gaze at the fixation cross, that was positioned straight ahead and was visible for the whole duration of the test.

For the running task, participants were asked to run at 10 km/h for 2 min in front of the black screen focusing on the sensations perceived while running. The participants wore a soundproof headset to prevent them from using the sound of the treadmill as an information about the running speed. The treadmill display was covered so that participants did not have any feedback on their actual running speed. After 2 min of running at 10 km/h, the participants were asked to pause for 1 min. After this short pause, they started running at 7 km/h and had to manipulate the treadmill speed until they found a running speed that they perceived as matching the reference speed of 10 km/h. This task was proposed in four different blocks in which the participants were asked to regulate the speed of the treadmill to match the previous speed while being presented with:

•

No optical flow (noOF)

•

An optical flow matching the treadmill speed (matchOF)

•
A faster optical flow (+5 km/h) (fastOF)

•
A slower optical flow (-5 km/h) (slowOF).

The order of presentation of the conditions was randomly selected. Note that the fixation cross was positioned so that it corresponded to the focus of expansion of the visual scene in the conditions with optical flow.

The treadmill speed perceived by participants as matching the reference speed and the time necessary to define this speed were recorded at the end of each block.

In between blocks, participants were asked to walk for about 2 min.

Results

For each condition, data distribution was assessed using the Shapiro-Wilk test (please refer to Supplementary Materials S1-T1 for the data). We then compared the selected running speed to the reference running speed using in each case either a t-test when data was normally distributed or a Wilcoxon signed rank test. For all four tests, the alpha level was corrected for multiple comparisons using Bonferroni correction (i.e., 0.05/4 = 0.0125). For all conditions, the selected running speed was not significantly different from the reference running speed after correction.

We then tested whether the selected running speed was different based on the optical flow condition. Since data was not normally distributed, a Friedman rank sum test for repeated measures was used. No differences were found for the selected running speed in the four conditions [𝜒 # (3) = 3.1, 𝑝 = 0.3765] (see Figure 2). Because we did not find any significant difference between the means (i.e., we could not reject H0), we also computed the Bayes factor. The Bayes factor was 0.37, a value which can be interpreted as an 'anecdotal' evidence in favour of H0 [START_REF] Van Doorn | The JASP Guidelines for Conducting and Reporting a Bayesian Analysis[END_REF]. Lastly, we tested whether the duration of the selection phase was different based on the optical flow condition. Data was not normally distributed, so a Friedman rank sum test for repeated measures was used. Here again, we did not find any significant difference between optical flow conditions regarding the duration of the running phase [𝜒 # (3) = 1.084, 𝑝 = 0.7809] (see Figure 3). As for the previous analysis, we computed the Bayes factor, which was 0.11, thereby constituting a moderate to strong evidence in favour of H0 [START_REF] Van Doorn | The JASP Guidelines for Conducting and Reporting a Bayesian Analysis[END_REF].

Discussion

Participants running on a treadmill were asked to match their running speed to a previously experienced reference running speed while being presented with different optical flow conditions. For all optical flow conditions, participants selected a running speed that did not differ significantly from the reference speed, and there was no difference in the selected speed between the four optical flow conditions. Also, the duration of the selection task did not differ significantly between the four optical flow conditions.

Previous studies with walking participants showed an influence of the optical flow rate on the control of locomotor speed [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF][START_REF] Pailhous | Unintentional modulations of human gait by optical flow[END_REF][START_REF] Powell | Optic flow with a stereoscopic display: Sustained influence on speed of locomotion[END_REF], suggesting an internal calibration between locomotion and visual perception. Here, the adaptation of running speed to the presented optical flow speed was not possible, as the treadmill was not self-driven, but should have influenced the ability to retrieve the reference running speed due to the disruption of this internal calibration. Our results show that this is not the case for running individuals. Moreover, for walking individuals, previous studies suggested the presence of a predetermined expectation of the visual effect that should be associated with a certain walking speed [START_REF] Durgin | Perception of visual speed while moving[END_REF]. This should also have led to expecting an influence on the ability to retrieve an imposed speed in the different optical flow conditions. It seems that visual incongruences do not have the same influence on running speed control as they do on walking speed control, possibly due to the higher automatization of running compared to walking as well as to the fact that it requires less cortical control [START_REF] Jahn | Brain activation patterns during imagined stance and locomotion in functional magnetic resonance imaging[END_REF][START_REF] Brandt | You are better off running than walking with acute vestibulopathy[END_REF].

Note that our results might be consistent with a statistically optimal integration model [START_REF] Ernst | Humans integrate visual and haptic information in a statistically optimal fashion[END_REF], which predicts that sensory information from multiple sources is weighted according to the estimation of the reliability of each sensory source. In our task, because running at the reference speed was performed without optical flow, participants might have given more weight to proprioceptive information than to visual information also in the second part of the running task.

Experiment 2

Introduction

In this experiment, we tested if treadmill running is influenced by the presence and the manipulation of an optical flow. Studies show that the manipulation of the optical flow has a destabilizing effect on postural stability [START_REF] Keshner | The influence of an immersive virtual environment on the segmental organization of postural stabilizing responses[END_REF][START_REF] Wright | Using virtual reality to induce cross-axis adaptation of postural control: Implications for rehabilitation[END_REF][START_REF] Wright | Using virtual reality to augment perception, enhance sensorimotor adaptation, and change our minds[END_REF][START_REF] Slobounov | Incorporating virtual reality graphics with brain imaging for assessment of sport-related concussions[END_REF], and a significant influence on locomotor patterns when walking [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF][START_REF] Pailhous | Unintentional modulations of human gait by optical flow[END_REF][START_REF] Baumberger | Walking in an environment of moving ground texture[END_REF][START_REF] Zijlstra | Voluntary and involuntary adaptation of walking to temporal and spatial constraints[END_REF]. The results regarding modifications of gait parameters and their variability as an effect of a virtual optical flow are contradicting, suggesting a great influence of the specific setup, the walking mode (i.e., fixed speed vs. self-paced treadmill) and the time to adapt to the new environment. Specifically, some studies found a lower stride length, with an increased step width [START_REF] Hollman | Does walking in a virtual environment induce unstable gait? An examination of vertical ground reaction forces[END_REF][START_REF] Hollman | Spatiotemporal gait deviations in a virtual reality environment[END_REF], an increased walking speed variability, and an increased step width variability [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF][START_REF] Hollman | Spatiotemporal gait deviations in a virtual reality environment[END_REF][START_REF] Kastavelis | The effect of virtual reality on gait variability[END_REF]. On the other hand, other authors found a stabilizing effect on locomotion patterns [START_REF] Guerin | Optical modulation of locomotion and energy expenditure at preferred transition speed[END_REF].

Here we investigated if the presence and manipulation of an optical flow could influence spatiotemporal gait parameters during treadmill running.

Methods

Participants

20 healthy participants (13 female, 7 male) with a mean age of 25.4 (±2.7 SD) participated in this study. They were unaware of the purpose of the study, were moderately trained to trained, had normal or corrected-to-normal vision and none had a history of cardiovascular disease. All participants gave their informed and written consent prior to the inclusion in the study. This study was performed in accordance with the ethical standards specified by the 1964 Declaration of Helsinki and approved by the Ethics Committee of the University of Fribourg.

Experimental Setup

The experimental setup was the same as in Experiment 1, except for the position of the treadmill, which was at a distance of 2.5 m from the screen, leading to an effective field of view of 70°.

While running, participants' heart rate (HR) was constantly monitored (Polar Team2 System, Polar Electro Oy-Kempele, Finland). Moreover, the NaturalPoint Optitrack system (NaturalPoint, Inc. DBA OptiTrack-Corvallis, OR, USA) with 16 OptiTrack cameras (Prime 17W model) was used to monitor the position of the markers that were used for the estimation of the spatiotemporal gait parameters.

Procedure

As in Experiment 1, prior to the running task, the participants filled out a custom-made questionnaire about their sports activity. For this experiment, also the short version of the International Physical Activity Questionnaire (IPAQ-French version) was used.

At the beginning of the experiment and prior to the testing phase, the participants spent 4 min familiarizing themselves with treadmill running. During this phase, they were instructed to "select an exercise intensity that you prefer and can be sustained for 20 min" [START_REF] Parfitt | The effect of prescribed and preferred intensity exercise on psychological affect and the influence of baseline measures of affect[END_REF][START_REF] Parfitt | The psychological and physiological responses of sedentary individuals to prescribed and preferred intensity exercise[END_REF]. The selected running speed was then used for the subsequent testing phase. To ensure that the participant's speed selection was solely based on perception, the speed display on the treadmill was covered so that it was only visible to the investigator. Self-selected running speed was used to reduce inter-individual differences, because fitness level and running experience have an influence on the parameters measured for this study.

The first part of the experiment allowed us to assess the visual speed perceived as matching the chosen running speed for each participant. Participants were presented with the visual scene while running at their preferred running speed. They were instructed to estimate if the optical flow was slower or faster than the actual running speed. Four consecutive tests were proposed in a random order using a one up-one down staircase method [START_REF] Kingdom | Psychophysics: A Practical Introduction[END_REF][START_REF] Leek | Adaptive procedures in psychophysical research[END_REF]. The optical flow started two times at a higher speed than the actual treadmill speed (+4 km/h) and two times at a lower speed (-4 km/h). The order of starting speed was randomized. The speed of the visual scene was adjusted according to the one up-one down staircase method, with an increase/decrease of the visual speed of 0.5 km/h until the first inversion of the participant's response, followed by steps of 0.3 km/h. This allowed us to determine the perceptual threshold that indicated the visual speed perceived as matching the actual running speed, i.e., the point of subjective equality (PSE). Prior to the task, participants were familiarized with the experimental setting and task using a training program. The single trials started with the participant running at the previously chosen speed. The visual scene was presented for 2 s before the participants were challenged with a black screen presenting the question "up or down", thus asking them to decide if they wanted to increase (i.e., up) or decrease (i.e., down) the speed of the visual scene. For each trial, participants gave their response while continuing their treadmill run by pressing a switch put on top of a light custom-made plastic cylinder (115 × 30 mm, 15 g) they were holding in each hand. The responses (i.e., left for up and right for down) were sent to the computer via Bluetooth and directly integrated into the staircase test. Once the participant pressed the chosen button, the graphics returned to the visual scene of the following trial (see Figure 4). Each staircase ended when 15 inversions of the responses were reached and the participants were free to take a pause before starting the subsequent test. The second part of the experiment consisted of 4 different blocks in which the participants were asked to run at constant speed for 3 min and 30 s while being presented with different optical flow conditions. The reference value for the optical flow speed was personalized for each participant using the PSE found with the staircase test. The four conditions were:

• no optical flow (noOF) • matching optical flow with a visual speed corresponding to the PSE (matchOF) • faster optical flow with a visual speed that was 40% higher than the PSE (fastOF) • slower optical flow with a visual speed that was 40% lower than the PSE (slowOF).

The order of conditions was randomly assigned. At the end of each block, the participants took a 2-min pause before starting the next block.

For all the parts involving the presentation of a visual speed, the participants were instructed to gaze at the fixation cross that was visible in all conditions.

At the end of the running phase, the participants estimated their perceived exertion using the Borg RPE Scale (6 to 20 scale), with values ranging from 'no exertion at all' to 'maximal exertion'.

Spatiotemporal Parameter Analysis

For the spatiotemporal gait parameter estimation, the information derived from the OptiTrack system was used. The participants were equipped with two reflective markers, one positioned on each heel. The data analysis was performed on steady gait by removing the first and the last 15 s of each block.

The different gait parameters were estimated using the heel marker trajectory and were averaged for each block, as described in Dubois and Bresciani [START_REF] Dubois | Validation of an ambient system for the measurement of gait parameters[END_REF]. Table 1 shows which spatiotemporal gait parameters were estimated and how they were calculated. The distance on the x-axis between the left and right heel For analysis, these values were averaged both for the whole duration of the block as also for each minute of each block.

Results

We first compared the perceived visual speed with the actual running speed using a paired ttest, as both running and perceived visual speed were normally distributed (please refer to Supplementary Materials S1-T2 for the data.) This allowed us to determine how visual speed was estimated compared to the actual running speed, i.e., treadmill speed. Participants set visual speed significantly higher than the actual treadmill speed (t(19) = -4.8743, p < 0.001). Figure 5 shows the difference between running speed and perceived visual speed. We then tested whether the presentation and manipulation of the optical flow affected spatiotemporal parameters in treadmill running using a 4x3 (Condition[noOF, matchOF, fastOF, slowOF]*minute [START_REF] Sun | Multisensory integration in the estimation of relative path length[END_REF][START_REF] Mergner | Interaction of vestibular, somatosensory and visual signals for postural control and motion perception under terrestrial and microgravity conditions-A conceptual model[END_REF][START_REF] Dietz | Proprioception and locomotor disorders[END_REF]) mixed analysis of variance (ANOVA). The results showed no main effect of the condition for all the spatiotemporal parameters, i.e., step length, step duration, step frequency, and step width. Even if there was no influence of the condition, the results showed a significant effect of the minute of the acquisition. There was no interaction between the main factors. A Tukey's post-hoc test indicated that the difference was significant when comparing minute 1 to minute 3 for step duration (p < 0.05), step frequency (p < 0.05) and step width (p < 0.01), while there was only a tendency for step length (p = 0.061).

For the HR, since a plateau was observed in all participants, the mean value was calculated as average of 30 s once the plateau was reached. A repeated-measures ANOVA showed a significant difference for mean HR [F(3,57) = 5.7465, p < 0.01], while the starting HR did not differ between conditions [χ # (3) = 0.6528, 𝑝 = 0.8842]. A Tukey's post-hoc test indicated a significant difference between the noOF and the VR conditions (noOF-matchOF: p < 0.01, noOF-fastOF: p < 0.05, noOF-slowOF: p < 0.001) as shown in Figure 6.

Discussion

First, participants running on a treadmill while being presented with a moving virtual scene were asked to match the visual speed of the VR scene to their actual running speed. Participants set visual speed significantly higher than the actual running speed, i.e., treadmill speed. In other words, the visual scene had to move significantly faster than the treadmill speed for the two speeds to be perceived as equivalent, indicating and underestimation of visual speed relative to treadmill speed. Then, spatiotemporal gait parameters and HR were measured with participants running on a treadmill at their preferred running speed while being presented with different optical flow conditions. At the end of each condition, perceived effort was recorded. We did not observe any significant difference between optical flow conditions regarding the spatiotemporal gait parameters. However, we observed an effect of the time (i.e., minute) of acquisition. In addition, the HR was significantly different in the noOF condition than in the VR conditions, while there was no difference between conditions regarding perceived effort.

Our results confirm the general tendency to underestimate visual speed relative to locomotor speed in virtual environments, both when walking [START_REF] Durgin | The perception of linear self-motion[END_REF][START_REF] Banton | The perception of walking speed in a virtual environment[END_REF][START_REF] Powell | Blurring the boundaries: The perception of visual gain in treadmill-mediated virtual environments[END_REF][START_REF] Kassler | Matching actual treadmill walking speed and visually perceived walking speed in a projection virtual environment[END_REF] and running [START_REF] Caramenti | Matching optical flow to motor speed in virtual reality while running on a treadmill[END_REF][START_REF] Caramenti | Regular physical activity modulates perceived visual speed when running in treadmill-mediated virtual environments[END_REF][START_REF] Caramenti | Influence of the size of the field of view on visual perception while running in a treadmill-mediated virtual environment[END_REF]. Compared to previous work with running participants, our results show that visual speed underestimation relative to running speed also occurs with participants running at their preferred running speed. Specifically, previous work was always based on imposed running speeds [START_REF] Caramenti | Matching optical flow to motor speed in virtual reality while running on a treadmill[END_REF][START_REF] Caramenti | Regular physical activity modulates perceived visual speed when running in treadmill-mediated virtual environments[END_REF][START_REF] Caramenti | Influence of the size of the field of view on visual perception while running in a treadmill-mediated virtual environment[END_REF], which were not necessarily in the range of preferred speeds of all participants. Here, every participant selected his/her running speed, reducing a possible influence of fatigue. This likely also reduced the attentional demand and fear of falling off of the treadmill [START_REF] Abernethy | The attentional demands of preferred and non-preferred gait patterns[END_REF] thanks to an increased felt safety compared to an imposed running speed.

Previous studies on walking showed that the manipulation of the optical flow can significantly affect locomotor patterns [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF][START_REF] Pailhous | Unintentional modulations of human gait by optical flow[END_REF][START_REF] Baumberger | Walking in an environment of moving ground texture[END_REF][START_REF] Zijlstra | Voluntary and involuntary adaptation of walking to temporal and spatial constraints[END_REF]. In particular, optical flow can modulate parameters known to be reflective of gait instability, such as step width, step length, stride cycle, and stride velocity variability [START_REF] Menz | Age-related differences in walking stability[END_REF][START_REF] Krebs | Is base of support greater in unsteady gait?[END_REF]. Our results show that this influence on locomotor patterns cannot be extended to individuals running on a treadmill. In fact, none of the spatiotemporal gait parameters (i.e., step length, step duration, step frequency, and step width) was significantly affected by the optical flow condition. This suggests that when running at a self-selected speed, the presence and manipulation of the optical flow has no stabilizing or destabilizing effect. The difference between our results and those previously reported with walking participants could be linked to the fact that running seems to be characterized by a higher automatization and a lower cortical control compared to walking [START_REF] Jahn | Brain activation patterns during imagined stance and locomotion in functional magnetic resonance imaging[END_REF][START_REF] Brandt | You are better off running than walking with acute vestibulopathy[END_REF]. The observed effect of the minute of acquisition could be linked to treadmill running accommodation, which can be different from individual to individual [START_REF] Nigg | A kinematic comparison of overground and treadmill running[END_REF]. Moreover, our results show that the presence and manipulation of the optical flow does not influence this accommodation process, since no difference was found between the four optical flow conditions.

Even if there was no significant effect of the optical flow conditions on spatiotemporal running parameters, VR seems to influence HR. Specifically, we found significant differences between the noOF and the VR conditions. Because the order of presentation of the conditions was randomized, exercise intensity was the same for each subject in all conditions, and starting HR did not differ between blocks. Therefore, the observed difference should not be linked to the running task per se. The higher HR values in the VR conditions may therefore be linked to an emotional activation, HR being a possible indicator of excitement [START_REF] Obrist | Cardiovascular Psychophysiology: A Perspective[END_REF]. This would also explain the absence of difference in perceived effort as measured with the Borg RPE Scale, even with the higher HR values in the VR conditions. In fact, even if there was no significant difference, there was a slight tendency toward lower RPE values for the VR conditions compared to the noOF condition. This could confirm the results of previous studies that suggest that VR exercise equipment could act as a distraction from the exercise intensity [START_REF] Annesi | Effects of virtual reality-enhanced exercise equipment on adherence and exerciseinduced feeling states[END_REF][START_REF] Mestre | Does virtual reality enhance exercise performance, enjoyment, and dissociation? An exploratory study on a stationary bike apparatus[END_REF].

General Discussion

The control of locomotion by the central nervous system relies on the integration of sensory information provided by different systems such as the visual, vestibular, proprioceptive, and auditory systems. When the information provided by one of these sensory inputs is considered as less reliable or is in conflict with other cues, the central nervous system tends to weight sensory information taking into account its estimated reliability. Our results suggest that when running in a treadmill-mediated virtual environment, participants tend to provide a higher weight to proprioceptive information than to visual information. Specifically, the presence and manipulation of a simulated optical flow did not affect the ability of participants to retrieve a 'proprioceptive' reference speed previously experienced. In addition, optical flow did not affect the spatiotemporal gait parameters known to be indicators of gait stability.

The results of Experiment 2 confirm that the speed of an artificial optical flow is underestimated relative to treadmill speed when running in treadmill-mediated virtual environments. In particular, our results show that this relative underestimation applies not only to situations in which the running speed is imposed, as shown in previous studies [START_REF] Caramenti | Matching optical flow to motor speed in virtual reality while running on a treadmill[END_REF][START_REF] Caramenti | Regular physical activity modulates perceived visual speed when running in treadmill-mediated virtual environments[END_REF][START_REF] Caramenti | Influence of the size of the field of view on visual perception while running in a treadmill-mediated virtual environment[END_REF], but also to situations in which participants are allowed to run at their preferred running speed. Specifically, in our study, the participants were free to select their running speed for the tests. This likely increased their feeling of safety compared to tests performed at an imposed running speed. It also likely lowered the influence of fatigue and the fear of falling off of the treadmill [START_REF] Abernethy | The attentional demands of preferred and non-preferred gait patterns[END_REF]. This means that the visual underestimation previously reported with participants running in virtual environments still applies at running speeds for which past experience from practice could have been used for recalibration [START_REF] Bingham | The rate of adaptation to displacement prisms remains constant despite acquisition of rapid calibration[END_REF][START_REF] Redding | Prism adaptation during target pointing from visible and nonvisible starting locations[END_REF]. This could be linked to treadmill running in itself, since it has been shown to influence speed perception. In fact, treadmill running has been shown to be perceived as faster compared to the same overground running speed [START_REF] Kong | Perception of self-selected running speed is influenced by the treadmill but not footwear[END_REF][START_REF] Kong | Unmatched perception of speed when running overground and on a treadmill[END_REF][START_REF] Marsh | Effect of treadmill and overground walking on function and attitudes in older adults[END_REF][START_REF] White | Comparison of vertical ground reaction forces during overground and treadmill walking[END_REF]. It has also been shown to influence the capacity to discriminate actual running speed due to differences in kinetics and kinematics compared to overground running [START_REF] Nigg | A kinematic comparison of overground and treadmill running[END_REF][START_REF] Riley | A kinematics and kinetic comparison of overground and treadmill running[END_REF]. Another factor that could have influenced the results is the specific experimental setup. In fact, studies suggest that visual speed is perceived more accurately as peripheral flow increases [START_REF] Banton | The perception of walking speed in a virtual environment[END_REF][START_REF] Caramenti | Influence of the size of the field of view on visual perception while running in a treadmill-mediated virtual environment[END_REF][START_REF] Thurrell | The influence of non-visual signals of walking on the perceived speed of optic flow[END_REF][START_REF] Thurrell | Reduction of perceived visual speed during walking: Effect dependent upon stimulus similarity to the visual consequences of locomotion[END_REF][START_REF] Nilsson | Establishing the range of perceptually natural visual walking speeds for virtual walking-in-place locomotion[END_REF]. Moreover, a 3D stereoscopic environment would have been more immersive than the simple 2D on the projection screen and studies show that 3D environments induce a stronger sense of presence during spatial navigation tasks [START_REF] Kober | Cortical correlate of spatial presence in 2D and 3D interactive virtual reality: An EEG study[END_REF]. Nevertheless, here we used a 2D setup since more immersive 3D virtual reality situations inducing a strong sense of presence have been shown to have a greater capability of inducing postural instability compared to 2D VR situations [START_REF] Slobounov | Modulation of cortical activity in 2D versus 3D virtual reality environments: An EEG study[END_REF] presenting also a greater possibility that participants might have experienced VR induced symptoms and effects [START_REF] Sharples | Virtual reality induced symptoms and effects (VRISE): Comparison of head mounted display (HMD), desktop and projection display systems[END_REF].

Though our study confirms the tendency to misperceive the speed of a simulated optical flow when running in treadmill-mediated virtual environments, it shows that this misperception might not have an influence on the control of treadmill running itself. This result suggests that the findings of previous studies that reported an influence of the optical flow rate on the control of locomotor speed might not apply when running on a treadmill. This difference between previous results and ours could result from a higher automatization of running compared to walking, with the former requiring less cortical control [START_REF] Jahn | Brain activation patterns during imagined stance and locomotion in functional magnetic resonance imaging[END_REF][START_REF] Brandt | You are better off running than walking with acute vestibulopathy[END_REF]. This would in turn lead to a reduced influence of visual incongruences on the control of running speed. This would be in line with the results of Jahn et al. [START_REF] Jahn | Visually induced gait deviations during different locomotion speeds[END_REF][START_REF] Jahn | Differential effects of vestibular stimulation on walking and running[END_REF], who found larger gait deviations with vestibular and visual stimulation during walking compared to running. Those results can probably be attributed to a suppression of sensory signals during unhindered running in order to avoid potential disturbances of the optimized spinal program controlling running motor pattern [START_REF] Armstrong | The supraspinal control of mammalian locomotion[END_REF][START_REF] Dietz | Spinal cord pattern generators for locomotion[END_REF]. In fact, studies have shown that locomotion without disturbances or obstacles is more independent of cortical control, the latter being mainly required when exact foot placement is needed, or when the system must react to external perturbations, notably by changing muscle activity to modify limb trajectories to step over obstacles [START_REF] Drew | Cortical and brainstem control of locomotion[END_REF]. This would also explain the lower impact of visual incongruences on balance and locomotor patterns while treadmill running, as shown in Experiment 2. Our results contradict previous studies on walking, which showed an influence of an artificial optical flow on postural stability [START_REF] Keshner | The influence of an immersive virtual environment on the segmental organization of postural stabilizing responses[END_REF][START_REF] Wright | Using virtual reality to induce cross-axis adaptation of postural control: Implications for rehabilitation[END_REF][START_REF] Wright | Using virtual reality to augment perception, enhance sensorimotor adaptation, and change our minds[END_REF][START_REF] Slobounov | Incorporating virtual reality graphics with brain imaging for assessment of sport-related concussions[END_REF] and on locomotor patterns. In particular, Pailhous et al. [START_REF] Pailhous | Unintentional modulations of human gait by optical flow[END_REF] found that when walking overground, stride length decreases in presence of an artificial optical flow. Prokop et al. [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF] found that when walking on a treadmill at constant speed, stride-cycle variability increases and the optical flow significantly modulates walking velocity. Hollman et al. [START_REF] Hollman | Spatiotemporal gait deviations in a virtual reality environment[END_REF] found that when walking at pre-determined speeds on a treadmill in VR, step width increases and step length decreases, whereas the variability of stride velocity and step width increases. All these parameters are known to characterize gait instability [START_REF] Menz | Age-related differences in walking stability[END_REF][START_REF] Krebs | Is base of support greater in unsteady gait?[END_REF], suggesting that the presence of VR during treadmill locomotion may lead to a more instable gait. In fact, a shorter step length and an increased step width are associated with a more conservative gait pattern, often adopted in case of instability or fear of falling. Other studies show that this influence on locomotor patterns extends also to the walk-run and run-walk transition, leading to changes in the transition speed based on the speed of the optical flow [START_REF] Guerin | Optical modulation of locomotion and energy expenditure at preferred transition speed[END_REF][START_REF] Mohler | Visual flow influences gait transition speed and preferred walking speed[END_REF]. Our results show that this influence on locomotor patterns cannot be extended to running individuals, and that for running at a self-selected speed, the presence and manipulation of the optical flow has no stabilizing or destabilizing effect. This could also be due to the fact that when visual cues are in conflict with other sensory information about self-motion, which is the case when manipulating optical flow speed without having the possibility to vary locomotor speed, visual information ceases to be dominant compared to other types of information [START_REF] Sun | Multisensory integration in speed estimation during self-motion[END_REF][START_REF] Harris | Visual and non-visual cues in the perception of linear self motion[END_REF]. In fact, studies show that when participants are asked to walk at a constant speed, they tend to weight information from proprioceptive cues higher than cues derived from the visual system [START_REF] Prokop | Visual influence on human locomotion modulation to changes in optic flow[END_REF][START_REF] Varraine | Interaction between different sensory cues in the control of human gait[END_REF][START_REF] Pailhous | Unintentional modulations of human gait by optical flow[END_REF].

Our results also indicate that the simulated optical flow does not influence the process of accommodation to treadmill running, because no differences were found between the four optical flow conditions. This accommodation process has been shown to vary between individuals [START_REF] Nigg | A kinematic comparison of overground and treadmill running[END_REF]. Healthy adults devoid of experience with treadmill running can familiarize themselves with the task after 6 min in a single session, and angular kinematics as well as spatiotemporal parameters are highly reliable after 2 min [START_REF] Lavcanska | Familiarization to treadmill running in young unimpaired adults[END_REF]. White et al. [START_REF] White | Within-day accommodation effects on vertical reaction forces for treadmill running[END_REF] found that there was no significant difference in vertical force after 30 s of treadmill running. The differences we found between minute 1 and 3 could be linked to the new accommodation required after each pause in between the blocks. This would be in line with the fact that previous studies have shown that minimal amounts of treadmill training are necessary before full accommodation to treadmill locomotion [START_REF] Schieb | Kinematic accommodation of novice treadmill runners[END_REF].

Surprisingly, our results suggest that the presentation of an artificial optical flow can influence HR. Specifically, the results of Experiment 2 show a significant difference between the HR monitored in the noOF condition and the HR monitored in the VR conditions. Because the HR is an indicator of excitement [START_REF] Obrist | Cardiovascular Psychophysiology: A Perspective[END_REF], the observed results could be linked to an emotional activation, which would also explain the absence of difference between the four optical flow conditions regarding perceived effort. This excitement could be due to the novelty of the virtual scene and to some sense of presence deriving from the optical flow compared to the traditional treadmill-running settings, which are often considered as boring [START_REF] Thompson Coon | Does participating in physical activity in outdoor natural environments have a greater effect on physical and mental wellbeing than physical activity indoors? A systematic review[END_REF]. This would be in line with previous studies showing that VR can act as a distraction from exercise intensity, favoring a dissociation of the attentional focus [START_REF] Annesi | Effects of virtual reality-enhanced exercise equipment on adherence and exerciseinduced feeling states[END_REF][START_REF] Mestre | Does virtual reality enhance exercise performance, enjoyment, and dissociation? An exploratory study on a stationary bike apparatus[END_REF], and improving some of the mood benefits of physical exercise [START_REF] Plante | Might virtual reality promote the mood benefits of exercise?[END_REF]. If confirmed, this effect could be exploited to increase engagement and adherence to physical activity, notably in the context of health and disease prevention.

Figure 1 .

 1 Figure 1. Experimental setup.

Figure 2 .

 2 Figure 2. Running speed selected as matching the reference running speed in different optical flow conditions. The bars represent the means and the error bars the 95% confidence interval of those means.

Figure 3 .

 3 Figure 3. Task duration. Time needed for the participants to retrieve the running speed they perceived as matching the previously experienced reference running speed. The bars represent the means and the error bars the 95% confidence interval of those means.

Figure 4 .

 4 Figure 4. Experimental design.

Figure 5 .

 5 Figure 5. Visual speed underestimation. Speed of the visual scene that was perceived as matching running/treadmill speed. The bars represent the means and the error bars the 95% confidence interval of those means.

Figure 6 .

 6 Figure 6. Mean heart rate. The bars represent the means and the error bars the 95% confidence interval of those means.

Table 1 .

 1 Gait parameters and their method of estimation

	Spatiotemporal Parameters	Estimation
	Step duration (s)	The duration between the local minima of the left and right heel
	Step length (m)	Treadmill speed (m/s) × Step duration (s)
	Step frequency (steps/s)	1/step duration
	Step width (cm)

Table 2

 2 reports all the results.

Table 2 .

 2 Results of the 4x3 mixed ANOVA. Significant difference: * = p<0.05, ** = p<0.01, *** = p<0.001.

	Variable	Condition	Minute	Interaction
	Step length	F(3,57) = 1.695, p = 0.178	F(2,38) = 10.000, p < 0.001 ***	F(6,114) = 1.423, p = 0.212
	Step duration	F(3,57) = 1.747, p = 0.168	F(2,38) = 10.085, p < 0.001 ***	F(6,114) = 1.357, p = 0.238
	Step frequency	F(3,57) = 1.977, p = 0.128	F(2,38) = 9.492, p < 0.001 ***	F(6,114) = 1.534, p = 0.173

Acknowledgments:

The authors thank Florian Blunschi and Lisa Perissinotto (Department of Neuroscience and Movement Science, University of Fribourg, Fribourg, Switzerland) for recruiting the participants and for providing support in data collection, Marco Scodeggio (Istituto di Bioimmagini e Fisiologia Molecolare, Consiglio Nazionale delle Ricerche, Segrate, Milano, Italy) for technical assistance in constructing the response device, and Thibaut Le Naour (Department of Neuroscience and Movement Science, University of Fribourg, Fribourg, Switzerland) for his help in the creation of the representation of the experimental set-up (Figure 1).

Conflicts of Interest:

The authors declare no conflict of interest.