Efficient Quantum Walk on a Quantum Processor

In this paper, we present efficient quantum circuits for implementing continuous time quantum walks (CTQWs) on circulant graphs with an eigenvalue spectrum that can be classically computed efficiently. These quantum circuits provide the time-evolution states of CTQWs on circulant graphs exponentially faster than best previously known methods ng2004iterative . We report a proof-of-principle experiment, where we implement CTQWs on an example circulant graph (namely the complete graph of four vertices) using a two-qubit photonics quantum processor to sample the probability distributions and perform state tomography on the output state of a CTQW. We also provide evidence from computational complexity theory that the probability distributions that are output from the circuits of this circulant form are hard to sample from using a classical computer, implying our scheme also provides an exponential speedup for sampling.

Efficient quantum circuit implementations of CTQWs have been presented for sparse and efficiently row-computable graphs aharonov2003adiabatic ; berry2007efficient , and specific non-sparse graphs childs2009limitations ; childs2010relationship . However, the design of quantum circuits for implementing CTQWs is in general difficult, since the time-evolution operator is time-dependent and non-local farhi1998quantum . A subset of circulant graphs have the property that their eigenvalues and eigenvectors can be classically computed efficiently gray2006toeplitz ; ng2004iterative . This enables us to construct a scheme that efficiently outputs the quantum state $\left|{\psi\left(t\right)}\right\rangle$, which corresponds to the time evolution state of a CTQW on corresponding graphs. One can then either perform direct measurements on $\left|{\psi\left(t\right)}\right\rangle$ or implement further quantum circuit operations to extract physically meaningful information. For example the “SWAP test” buhrman2001quantum can be used to estimate the similarity of dynamical behaviors of two circulant Hamiltonians operating on two different initial states, as shown in Figure 1(A). This procedure can also be adapted to study the stability of quantum dynamics of circulant molecules (for example, the DNA Möbius strips han2010folding ) in a perturbational environment peres1984stability ; prosen2002stability .

On the other hand, when measuring $\left|{\psi\left(t\right)}\right\rangle$ in the computational basis we can sample the probability distribution

that describes the probability of observing the quantum walker at position $x\in{\left\{{0,1}\right\}^{n}}$—an $n$-bit string, corresponding to the vertices of the given graph, as shown in Figure 1(B). Sampling of this form is sufficient to solve various search and characterization problems childs2004spatial ; sanchez2012quantum , and can be used to deduce critical parameters of the quantum walk, such as mixing time kempe2003quantum . It is unlikely for a classical computer to be able to efficiently sample from $p(x)$. We adapt the similar methodology of refs. aaronson2011computational ; bremner2010classical ; bremner2015average to show that if there did exist a classical sampler for a somewhat more general class of circuits, this would have the following unlikely complexity-theoretic implication: the infinite tower of complexity classes known as the polynomial hierarchy would collapse. This evidence of hardness exists despite the classical efficiency with which properties of the CTQW, such as the eigenvalues of circulant graphs, can be computed on a classical machine.

For an undirected graph $G$ of $N$ vertices, a quantum particle (or “quantum walker”) placed on $G$ evolves into a superposition $\left|\psi(t)\right\rangle$ of states in the orthonormal basis $\left\{{\left|1\right\rangle,\left|2\right\rangle,\dots,\left|N\right\rangle}\right\}$ that correspond to vertices of $G$. The exact evolution of the CTQW is governed by connections between the vertices of $G$: $\left|{\psi\left(t\right)}\right\rangle=\exp(-itH)\left|{\psi\left(0\right)}\right\rangle$ where the Hamiltonian is given by $H=\gamma A$ for hopping rate per edge per unit time $\gamma$ and where $A$ is the $N$-by-$N$ symmetric adjacency matrix, whose entries are ${A_{jk}}={\rm{}}1$, if vertices $j$ and $k$ are connected by an edge in $G$, and ${A_{jk}}={\rm{}}0$ otherwise farhi1998quantum .

Circulant graphs are defined by symmetric circulant adjacency matrices for which each row $j$ when right-rotated by one element, equals the next row $j+1$—for example complete graphs, cycle graphs and Mobius ladder graphs are all subclasses of circulant graphs. It follows that Hamiltonians for CTQWs on any circulant graph have a symmetric circulant matrix representation, which can be diagonalized by the unitary Fourier transform gray2006toeplitz , i.e. $H={Q^{\dagger}}\Lambda Q$, where

and $\Lambda$ is a diagonal matrix containing eigenvalues of $H$, which are all real and whose order is determined by the order of the eigenvectors in $Q$. Consequently, we have $\exp(-itH)={Q^{\dagger}}\exp(-it\Lambda)Q$, where the time dependence of $\exp(-itH)$ is confined to the diagonal unitary operator $D=\exp(-it\Lambda)$.

The Fourier transformation $Q$ can be implemented efficiently by the well-known QFT quantum circuit nielsen2010quantum . For a circulant graph that has $N=2^{n}$ vertices, the required QFT of $N$ dimension can be implemented with $O((\log{N})^{2})=O(n^{2})$ quantum gates acting on $O(n)$ qubits. To implement the inverse QFT, the same circuit is used in reverse order with phase gates of opposite sign. $D$ can be implemented using at most $N$ controlled-phase gates with phase values being a linear function of $t$, because an arbitrary phase can be applied to an arbitrary basis state, conditional on at most $n-1$ qubits. Given a circulant graph that has $O(\operatorname{poly}(n))$ non-zero eigenvalues, only $O(\operatorname{poly}(n))$ controlled-phase gates are needed to implement $D$. If the given circulant graph has $O(2^{n})$ distinct eigenvalues, which can be characterised efficiently (such as the cycle graphs and Mobius ladder graphs), we are still able to implement the diagonal unitary operator $D$ using polynomial quantum resources. A general construction of efficient quantum circuits for $D$ was given by Childs childs2004quantum , and is shown in the Appendix for completeness. Thus, the quantum circuit implementations of CTQWs on circulant graphs can be constructed, which have an overall complexity of $O(\operatorname{poly}(n))$, and act on at most $O(n)$ qubits. Compared with the best known classical algorithm based on fast Fourier transform, that has the computational complexity of $O(n2^{n})$ ng2004iterative , the proposed quantum circuit implementation generates the evolution state $\left|{\psi\left(t\right)}\right\rangle$ with an exponential advantage in speed.

Consider a circuit of the form $Q^{\dagger}DQ$, where $D$ is a diagonal matrix made up of $\operatorname{poly}(n)$ controlled-phase gates. Define

${H^{\otimes n}}D{H^{\otimes n}}$ represents a family of circuits having the following structure: each qubit line begins and ends with a Hadamard ($H$) gate, and, in between, every gate is diagonal in the computational basis. This class of circuits is known as instantaneous quantum polynomial time (IQP) shepherd2009temporally ; bremner2010classical . It is known that computing $p_{D}$ for arbitrary diagonal unitaries $D$ made up of circuits of $\operatorname{poly}(n)$ gates, even if each acts on $O(1)$ qubits, is #P-hard bremner2015average ; fujii13 ; goldberg14 . This hardness result even holds for approximating $p_{D}$ up to any multiplicative error strictly less than $1/2$ fujii13 ; goldberg14 , where $\widetilde{p_{D}}$ is said to approximate $p_{D}$ up to multiplicative error $\epsilon$ if

Towards a contradiction, assume that there exists a polynomial-time randomized classical algorithm which samples from $p$. Then a classic result of Stockmeyer stockmeyer1985on states that there is an algorithm in the complexity class FBPP${}^{\text{NP}}$ which can approximate any desired probability $p(x)$ to within multiplicative error $O(1/\operatorname{poly}(n))$. This complexity class FBPP${}^{\text{NP}}$—described as polynomial-time randomized classical computation equipped with an oracle to solve arbitrary NP problems—sits within the infinite tower of complexity classes known as the polynomial(-time) hierarchy papadimitriou1994computational . Combining with the above hardness result of approximating $p_{D}$, we find that the assumption implies that an FBPP${}^{\text{NP}}$ algorithm solves a #P-hard problem, so P${}^{\text{\#P}}$ would be contained within FBPP${}^{\text{NP}}$, and therefore the polynomial hierarchy would collapse to its third level. This consequence is considered very unlikely in computational complexity theory papadimitriou1994computational .

We therefore conclude that a polynomial-time randomized classical sampler from the distribution $p$ is unlikely to exist. Further, this even holds for classical algorithms which sample from any distribution $\widetilde{p}$ which approximates $p$ up to multiplicative error strictly less than $1/2$ in each probability $p(x)$. It is worth noting that if the output distribution results from measurements on only $O(\operatorname{poly}\log n)$ qubits van2011simulating , or obeys the sparsity promise that only a $\operatorname{poly}(n)$-sized, and a priori unknown, subset of the measurement probabilities are nonzero schwarz2013simulating , it could be classically efficiently sampled. The proof of hardness here does not hold for the approximate sampling from $p$ up to small additive error. It is an interesting open question whether similar techniques to aaronson2011computational ; bremner2015average can be used to prove hardness of the approximate case, perhaps conditioned on other conjectures in complexity theory.

As an experimental demonstration, we used a photonic quantum logic to simulate CTQWs on the $K_{4}$ graph—a complete graph with self loops on four vertices (Figure 2(A)). Complete graphs are a special kind of circulant graph, with an adjacency matrix $A$ where $A_{jk}=1$ for all $j,k$. The Hamiltonian of a complete graph on $N$ vertices has only 2 distinct eigenvalues, 0 and $N\gamma$. Therefore, the diagonal matrix of eigenvalues of $K_{4}$ is $\Lambda=\mbox{diag}(\{4\gamma,0,0,0\})$. Following the aforementioned discussion, we can readily construct the quantum circuit for implementing CTQWs on $K_{4}$ graph based on diagonalization using the QFT matrix. However, the choice of using the QFT matrix as the eigenbasis of Hamiltonian is not strictly necessary – any equivalent eigenbasis can be selected. Through the diagonalization using Hadamard eigenbasis, an alternative efficient quantum circuit for implementing CTQWs on $K_{4}$ graph is shown in Figure 2(B), which can be easily extended to the complete graph on $N$ vertices.

We built a configurable two-qubit photonics quantum processor (Figure 2(C)), adapting the entanglement-based technique presented in zhou2011adding , and implemented CTQWs on $K_{4}$ graph with various evolving times and initial states. Specifically, we prepared two different initial states ${\left|{{\varphi_{ini}}}\right\rangle_{1}}=\left[{1,0,0,0}\right]^{\prime}$ and ${\left|{{\varphi_{ini}}}\right\rangle_{2}}=\frac{1}{{\sqrt{2}}}\left[{1,1,0,0}\right]^{\prime}$, which represent the quantum walker starting from vertex 1, and the superposition of vertices 1 and 2 respectively. We chose the evolution time following the list $\left\{{0,\frac{1}{8}\pi,\frac{2}{8}\pi,\frac{3}{8}\pi,\frac{4}{8}\pi,\frac{5}{8}\pi,\frac{6}{8}\pi,\frac{7}{8}\pi,\pi}\right\}$, which covers the whole periodical characteristics of CTQWs on $K_{4}$ graph. For each evolution, we sampled the corresponding probability distribution with fixed integration time, shown in Figure 3(A) and (B). To measure how close the experimental and ideal probability distributions are, we calculated the average fidelities defined as ${F_{average}}=\frac{1}{9}\sum\nolimits_{n=1}^{9}{\sum\nolimits_{i=1}^{4}{\sqrt{{P_{ideal,n}}(i){P_{\exp,n}}(i)}}}$. The achieved average fidelities for the samplings with two distinct initial states are 96.68$\pm$0.27% and 95.82$\pm$0.25% respectively. Through the proposed circuit implementation, we are also able to examine the evolution states using quantum state tomography, which is generally difficult for the analog simulations. For two specific evolution states ${\left|{{\varphi_{out}}}\right\rangle_{1}}=\exp({-iH\frac{7}{8}\pi}){\left|{{\varphi_{ini}}}\right\rangle_{1}}$ and ${\left|{{\varphi_{out}}}\right\rangle_{2}}=\exp({-iH\frac{7}{8}\pi}){\left|{{\varphi_{ini}}}\right\rangle_{2}}$, we performed quantum state tomography and reconstructed the density matrices using the maximum likelihood estimation technique. The two reconstructed density matrices achieve fidelities of 85.81$\pm$1.08% and 88.44$\pm$0.97% respectively, shown in Figure 3(C) and (D).

In this paper, we have described how CTQWs on circulant graphs can be efficiently implemented on a quantum computer, if the eigenvalues of the graphs can be characterised efficiently classically. In fact, we can construct an efficient quantum circuit to implement CTQWs on any graph whose adjacency matrix is efficiently diagonalisable, in other words, as long as the matrix of column eigenvectors $Q$ and the diagonal matrix of the eigenvalue exponentials $D$ can be implemented efficiently. We have shown that the problem of sampling from the output probability distributions of quantum circuits of the form $Q^{\dagger}DQ$ is hard for classical computers, based on a highly plausible conjecture that the polynomial hierarchy does not collapse. This observation is particularly interesting from both perspectives of CTQW and computational complexity theory, as it provides new insights into the CTQW framework and also helps to classify and identify new problems in computational complexity theory. For the CTQWs on the circulant graphs of $\mbox{poly}(n)$ non-zero eigenvalues, the proposed quantum circuit implementations do not need a fully universal quantum computer, and thus can be viewed as an intermediate model of quantum computation. Although the hardness of the approximate case of the sampling problem is unknown, the evidence we provided for the exact case indicates a promising candidate for experimentally establishing quantum supremacy over classical computers, and further evidence against the extended Church-Turing thesis. Compared with other intermediate models such as the one clean qubit model knill1998power , IQP and boson sampling aaronson2011computational , the quantum circuit implementation of CTQWs is also more appealing due to available methods in fault tolerance and error correction, which are difficult to implement for other models rohde2012error . This may also lead onto other practical applications through the use of CTQWs for quantum algorithm design.

As shown in Figure 2(B), the quantum circuit for implementing CTQW on the $K_{4}$ graph consists of Hadamard gates (H), Pauli-X gates (X) and controlled-phase gate (CP). CP is implemented by configuring ${P_{1}}=\left|H\right\rangle\left\langle H\right|$, ${P_{2}}=I$, ${Q_{1}}=\left|V\right\rangle\left\langle V\right|$, ${Q_{2}}=R(=\left[{1,0;0,{e^{-i4\gamma t}}}\right])$, where $P_{1}$ and $Q_{1}$ are implemented by polarizers. Together with combining the operation $\left({H\cdot X}\right)\otimes\left({H\cdot X}\right)$ before CP with state preparation and the operation $\left({X\cdot H}\right)\otimes\left({X\cdot H}\right)$ after CP with measurement setting, we implement the whole quantum circuit on the experimental setup. The evolution time of CTQW is controlled by the phase value of $R$, which is determined by setting the three wave plates of $Q_{2}$ in Figure 2(C) to QWP($\frac{\pi}{4}$), HWP($\omega$), QWP($\frac{\pi}{4}$), where the angle $\omega$ of HWP equals to the phase $\theta$ of $R$. The evolution time $t$ is then given by ${{t=-\omega}\mathord{\left/{\vphantom{{t=-\omega}{\left({4\gamma}\right)}}}\right.\kern-1.2pt}{\left({4\gamma}\right)}}.$