# Efficient Implementation of Discrete-Time Quantum Walks on Quantum Computers

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## Abstract

**:**

## 1. Introduction

## 2. The Model: DTQW on the N-Cycle

^{⊺}denoting the transpose without complex conjugation; the position basis states are $|{j}_{p}{\rangle =(0,\dots ,0,1,0,\dots ,0)}^{\u22ba}$, with the only nonzero element in position j. Accordingly, a generic coin–position basis state $|{s}_{c}\rangle |{j}_{p}\rangle =|{s}_{c}\rangle \otimes |{j}_{p}\rangle $ is represented by the column vector of length $2N$, whose first N entries are related to $s=0$ and the last N to $s=1$. The only nonzero entry is the $(Ns+j)$-th one.

**Figure 1.**Schematic representation of a DTQW on the N-cycle. (

**a**) The coin state (internal degree of freedom) is responsible for making the walker move in the cycle clockwise if $|{1}_{c}\rangle $ and counterclockwise if $|{0}_{c}\rangle $. (

**b**) States and operators of a DTQW. The vertices of the cycle (light violet)—walker’s position states—are labeled by $|{j}_{p}\rangle $ with $j=0,1,\dots ,N-1$, and each vertex comprises two subvertices—coin states—labeled by $|{0}_{c}\rangle $ (orange) and $|{1}_{c}\rangle $ (green). Each step of the walk, Equation (2), involves the action of a local coin operator ${C}_{j}$ responsible for mixing the coin states of each vertex (we assume ${C}_{j}=C$$\forall j$), followed by the action of the conditional-shift operators $|{0}_{c}\rangle \langle {0}_{c}|\otimes {P}_{0}$ (decrement) and $|{1}_{c}\rangle \langle {1}_{c}|\otimes {P}_{1}$ (increment) responsible for shifting the position states, see Equation (3) [43].

## 3. Quantum Circuit Implementing the DTQW on the ${2}^{n}$-Cycle

#### 3.1. Quantum Circuit Design

#### 3.2. Comparison with Other Existing Schemes

## 4. Results and Discussion

#### 4.1. Hadamard DTQW

#### 4.2. Figures of Merit

#### 4.3. Circuit Optimization

#### 4.4. Analysis of the DTQW on the 4- and 8-Cycles

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CTQW | Continuous-time quantum walk |

DTQW | Discrete-time quantum walk |

(I)QFT | (Inverse) quantum Fourier transform |

## Appendix A. Eigenvalues and Eigenvectors of Circulant Matrices

^{⊺}denoting the transpose without complex conjugation. Letting $\Lambda =diag\left({\left\{{\lambda}_{m}\right\}}_{m}\right)$, we have

## Appendix B. Optimization of the Circuit for an Initially Localized Walker

## Appendix C. Analysis of the Size of DTQW Quantum Circuits in Different Schemes

- (i)
- The ID scheme here considered and the QFT scheme are based on the same circuit design (Figure 2d), in which the coin gate (one-qubit gate) cannot be executed in parallel to the CNOT gates, because it acts on their control qubit. Hence, the coin gate has unit depth and prevents the simplification of CNOT gates when iteratively concatenating the single time-step quantum circuit to obtain the successive steps. Therefore, in Appendix C.1 and Appendix C.2, the analysis focuses on a single time step, because the size of a circuit implementing t time steps is t times the size of that implementing a single time step.
- (ii)
- Controlled gates having different target qubits but the same control qubit (here, it is the coin) are executed in sequence (not in parallel) in actual quantum computers. This affects the depth of circuits and regards the CNOT gates in the ID and QFT schemes (Appendix C.1 and Appendix C.2, respectively) and the controlled-${R}_{k}$ gates in our scheme (Appendix C.3).
- (iii)
- The (I)QFT on an n-qubit register (no SWAP) requires n one-qubit gates (Hadamard) and $n(n-1)/2$ two-qubit gates (controlled-${R}_{k}$) and has depth $2n-1$ [86]. This regards the analysis of the QFT scheme (Appendix C.2) and our scheme (Appendix C.3).

#### Appendix C.1. ID Scheme

#### Appendix C.2. QFT Scheme

#### Appendix C.3. Present Scheme

## Appendix D. Transpilation of the Proposed Quantum Circuit for the Hadamard DTQW on the 4- and 8-Cycles

**Figure A1.**Gate count in the quantum circuits proposed in the present work for the Hadamard DTQW on the (

**a**) 4-cycle and (

**b**) 8-cycle (Section 4.1): depth $\mathcal{D}$ and number of one- and two-qubit gates, ${\mathcal{N}}^{\left(1\right)}$ and ${\mathcal{N}}^{\left(2\right)}$, respectively, as a function of time steps t with optimization_level=1,3 in transpilation (ibm_cairo). The quantum circuits for $t=1$ are shown in Figure A2 (4-cycle) and Figure A3 (8-cycle).

**Figure A2.**Quantum circuit proposed in the present work (Figure 4) for one step of the Hadamard DTQW (Section 4.1) on the 4-cycle transpiled in ibm_cairo with (

**a**) optimization_level=1 and (

**b**) optimization_level=3. Each qubit is labelled by the corresponding index in the qubit connectivity map in Figure 6a.

**Figure A3.**The same as in Figure A2, but for the 8-cycle with optimization_level=1.

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**Figure 2.**(

**a**) Quantum circuit implementing one time step of the DTQW on the ${2}^{n}$-cycle based on controlled-increment (I) and controlled-decrement (D) gates [48]. (

**b**) Increment and (

**c**) decrement gates consist of generalized CNOT gates, with controls being $|1\rangle $ (solid circle) and $|0\rangle $ (empty circle), respectively. These gates act on the walker’s position quantum register, conditional on the coin’s qubit state (see panel (

**a**)). (

**d**) The ID-quantum circuit shown in panel (

**a**) can be conveniently re-designed in terms of one increment gate (not controlled by the coin qubit) and CNOT gates only, being $\mathrm{Decr}.=\left({\u2a02}_{k=1}^{n}{X}_{k}\right)\mathrm{Incr}.\left({\u2a02}_{k=1}^{n}{X}_{k}\right)$ [58]. Quantum circuits in panels (

**a**,

**d**) implement the conditional shift operator $S={\sum}_{j=0}^{N-1}\left(\right|{0}_{c}\rangle \langle {0}_{c}|\otimes |{(j+1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}N)}_{p}\rangle \langle {j}_{p}|+|{1}_{c}\rangle \langle {1}_{c}|\otimes |{(j-1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}N)}_{p}\rangle \langle {j}_{p}\left|\right)$, having the opposite convention to (3) used in the present work.

**Figure 3.**Quantum circuit implementing one time step of the DTQW on the ${2}^{n}$-cycle proposed in [58]. The quantum Fourier transform, $\tilde{\mathcal{F}}$, and its inverse, ${\tilde{\mathcal{F}}}^{\u2020}$, do not include the SWAP gates. The increment gate is diagonalized by the QFT (see also Figure 2d). Conditional shift operator is as in Figure 2.

**Figure 4.**Quantum circuit implementing one time step of the DTQW on the ${2}^{n}$-cycle proposed in the present work. The quantum Fourier transform, $\tilde{\mathcal{F}}$, and its inverse, ${\tilde{\mathcal{F}}}^{\u2020}$, do not include the SWAP gates. To implement t time steps of the DTQW, we have to concatenate the above circuit t times. In doing so, since the QFT is unitary, ${\tilde{\mathcal{F}}}^{\u2020}\tilde{\mathcal{F}}=I$, we are left with only one QFT at the beginning, the central block (shaded) repeated t times, and one IQFT at the end. This simplification cannot occur in the quantum circuit in Figure 3 due to the CNOT and the coin gates. For an initially localized walker, $|{\psi}_{0}\rangle =|{\varphi}_{c}\rangle \otimes |{0}_{p}\rangle $, the initial QFT is conveniently replaced by a layer of Hadamard gates (see Appendix B).

**Figure 5.**Metrics in Table 1 as a function of the number of position qubits, n, and the number of time steps, t, of a DTQW on the ${2}^{n}$-cycle. Each column corresponds to a different scheme: (

**a**,

**e**,

**i**) Present, (

**b**,

**f**,

**j**) QFT, (

**c**,

**g**,

**k**) ID (lin.-depth), and (

**d**,

**h**,

**l**) ID (ancillae). Each row corresponds to a different metric: (

**a**–

**d**) Number of one-qubit gates ${\mathcal{N}}^{\left(1\right)}$, (

**e**–

**h**) Number of two-qubit gates ${\mathcal{N}}^{\left(2\right)}$, and (

**i**–

**l**) circuit depth $\mathcal{D}$.

**Figure 6.**(

**a**) Qubit connectivity map of ibm_cairo. (

**b**,

**c**) Optimal mapping of the coin–position state onto multiqubit state $|{q}_{n}^{c}{q}_{n-1}^{p}\dots {q}_{0}^{p}\rangle $ for the cycle with (

**b**) $N=4$ and (

**c**) $N=8$ vertices ($n=2,3$ position qubits, respectively). No SWAP operations between position and coin qubits are required by the controlled-${R}_{k}$ gates in Figure 4, the coin qubit (orange) being already adjacent to all position qubits (blue).

**Figure 7.**Hadamard DTQW on the $(N=4)$-cycle with initial state as in Equation (23). (

**a**) Hellinger fidelity between the ideal and the experimental probability distributions of walker’s position as a function of time step t. The experimental distributions include noisy simulation and implementations on the actual quantum hardware with optimization_level=1 (IBM Cairo) and optimization_level=3 (IBM Cairo Opt.) in transpilation. Results for the noisy simulation of the DTQW circuit in the QFT scheme [58] are reported for comparison. (

**b**) Ideal and experimental (IBM Cairo Opt.) probability distributions of walker’s position for time steps $t=0,\dots ,8$. Results for both simulations and quantum hardware implementation are obtained for ${10}^{5}$ shots and by encoding the position state in qubits 3 and 8 and the coin state in qubit 5 of ibm_cairo, see Figure 6a,b.

**Figure 9.**Recurrent generation of maximally entangled single-particle states for a Hadamard DTQW on the $(N=4)$-cycle with initial state as in Equation (23) investigated by means of the second-order Rényi entropy as a function of time steps t. Bipartite entanglement between the two parts (coin and position degrees of freedom) exists if the second-order Rényi entropy of a part is larger than that of the total system. Results of entropies are obtained via 300 randomized measurements [69] and ${10}^{5}$ shots for each step of the DTQW, with optimization_level=1 in transpilation. Position state is encoded in qubits 3 and 8 and the coin state in qubit 5 of ibm_cairo, see Figure 6a,b.

**Table 1.**Metrics of the quantum circuit implementing t time steps of a DTQW on the ${2}^{n}$-cycle for different schemes: ${\mathcal{N}}^{\left(1\right)}$ and ${\mathcal{N}}^{\left(2\right)}$ denote the number of one- and two-qubit gates, respectively, $\mathcal{D}$ the depth of the circuit, and ${\mathcal{N}}^{\left(a\right)}$ the number of ancilla qubits. The number n refers to the number of qubits encoding the walker’s position. See also Figure 5.

Scheme | Figure | One-Qubit ${\mathcal{N}}^{\left(1\right)}$ | Two-Qubit ${\mathcal{N}}^{\left(2\right)}$ | Depth $\mathcal{D}$ | Ancillae ${\mathcal{N}}^{\left(\mathit{a}\right)}$ |
---|---|---|---|---|---|

Present work | 4 | $t(n+1)+2n$ | $t(n-1)+n(n-1)$ | $tn+2(2n-1)$ | 0 |

QFT [58] | 3 | $t(3n+1)$ | $tn(n+1)$ | $t\left(6n\right)$ | 0 |

ID * [48] (lin.-depth q.c. [61], $n\ge 3$) | 2d | $2t$ | $t\frac{1}{3}(2{n}^{3}-6{n}^{2}+13n-3)$ | $t(4{n}^{2}-14n+19)$ | 0 |

ID * [48] (ancillae [62], $n\ge 4$) | 2d | $2t$ | $t(10{n}^{2}-48n+66)$ | $t(8{n}^{2}-38n+55)$ | $n-3$ |

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**MDPI and ACS Style**

Razzoli, L.; Cenedese, G.; Bondani, M.; Benenti, G.
Efficient Implementation of Discrete-Time Quantum Walks on Quantum Computers. *Entropy* **2024**, *26*, 313.
https://doi.org/10.3390/e26040313

**AMA Style**

Razzoli L, Cenedese G, Bondani M, Benenti G.
Efficient Implementation of Discrete-Time Quantum Walks on Quantum Computers. *Entropy*. 2024; 26(4):313.
https://doi.org/10.3390/e26040313

**Chicago/Turabian Style**

Razzoli, Luca, Gabriele Cenedese, Maria Bondani, and Giuliano Benenti.
2024. "Efficient Implementation of Discrete-Time Quantum Walks on Quantum Computers" *Entropy* 26, no. 4: 313.
https://doi.org/10.3390/e26040313