Sampling from the stationary distribution is one of the fundamental tasks of Markov-chain-based algorithms and has important applications in machine learning, combinatorial optimization, and network science. For the quantum case, quantum sampling from Markov chains corresponds to preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution instead of classical sampling from a stationary distribution. A different quantum sampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks. We build a quantum sampling algorithm that not only accelerates nonregular graphs, but also keeps the speed up of existing quantum algorithms for regular graphs. In nonregular graphs, the invocation of the quantum fast-forwarding algorithm accelerates previous state-of-the-art quantum sampling algorithms for both discrete-time and continuous-time cases, especially on sparse graphs. Compared to existing algorithms, we decrease the runtime by a factor of $logn$, where $n$ is the number of graph vertices. In regular graphs, our results match other quantum algorithms and the reliance on the gap of Markov chains achieves quadratic speedup compared with classical cases. For both cases, we reduce the number of ancilla qubits required compared with the existing results. In some widely used graphs and a series of sparse graphs where stationary distributions are difficult to reach quickly, our algorithm achieves complete quadratic acceleration (without log factor) over the classical case without any limit. In addition, we construct a different reflection around the stationary state with fewer ancilla qubits and believe it may have an independent application.
Sampling from the stationary distribution is one of the fundamental tasks of Markov-chain-based algorithms and has important applications in machine learning, combinatorial optimization, and network science. For the quantum case, quantum sampling from Markov chains corresponds to preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution instead of classical sampling from a stationary distribution. A different quantum sampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks. We build a quantum sampling algorithm that not only accelerates nonregular graphs, but also keeps the speed up of existing quantum algorithms for regular graphs. In nonregular graphs, the invocation of the quantum fast-forwarding algorithm accelerates previous state-of-the-art quantum sampling algorithms for both discrete-time and continuous-time cases, especially on sparse graphs. Compared to existing algorithms, we decrease the runtime by a factor of logn, where n is the number of graph vertices. In regular graphs, our results match other quantum algorithms and the reliance on the gap of Markov chains achieves quadratic speedup compared with classical cases. For both cases, we reduce the number of ancilla qubits required compared with the existing results. In some widely used graphs and a series of sparse graphs where stationary distributions are difficult to reach quickly, our algorithm achieves complete quadratic acceleration (without log factor) over the classical case without any limit. In addition, we construct a different reflection around the stationary state with fewer ancilla qubits and believe it may have an independent application.
The problem of sampling from the stationary distribution of a Markov chain finds widespread applications in a variety of fields. The time required for a Markov chain to converge to its stationary distribution is known as the classical mixing time. In this article, we deal with analog quantum algorithms for mixing. First, we provide an analog quantum algorithm that, given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of the square root of the classical mixing time and the square root of the classical hitting time. Our algorithm makes use of the framework of interpolated quantum walks and relies on Hamiltonian evolution in conjunction with von Neumann measurements. There also exists a different notion for quantum mixing: the problem of sampling from the limiting distribution of quantum walks, defined in a time-averaged sense. In this scenario, the quantum mixing time is defined as the time required to sample from a distribution that is close to this limiting distribution. Recently, we provided an upper bound on the quantum mixing time for Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we also extend and expand upon our findings therein. Namely, we provide an intuitive understanding of the state-of-the-art random matrix theory tools used to derive our results. In particular, for our analysis we require information about macroscopic, mesoscopic, and microscopic statistics of eigenvalues of random matrices which we highlight here. Furthermore, we provide numerical simulations that corroborate our analytical findings and extend this notion of mixing from simple graphs to any ergodic, reversible, Markov chain.
We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with P the Markov chain transition matrix and D = root P omicron P-T its discriminant matrix (D = P if P is symmetric), we construct a quantum walk algorithm that for any quantum state vertical bar v > and integer t returns a quantum state epsilon-close to the state D-t vertical bar v > /parallel to D-t vertical bar v > parallel to. The algorithm uses O(parallel to D-t vertical bar v > parallel to(-1) root t log(epsilon parallel to D-t vertical bar v > parallel to(-1)) expected quantum walk steps and O(parallel to D-t vertical bar v > parallel to(-1)) expected reflections around vertical bar v >. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.
Quantum algorithms can potentially solve a handful of problems more efficiently than their classical counterparts. In that context, it has been discussed that Markov chains problems could be solved significantly faster using quantum computing. Indeed, previous work suggests that quantum computers could accelerate sampling from the stationary distribution of reversible Markov chains. However, in practice, certain physical processes of interest are nonreversible in the probabilistic sense and reversible Markov chains can sometimes be replaced by more efficient nonreversible chains targeting the same stationary distribution. This study constructs Markov chain reversibilizations and develops quantum algorithmic techniques to accelerate nonreversible processes. Such an up-to-exponential quantum speedup goes beyond the predicted quadratic quantum acceleration for reversible chains and is likely to have a decisive impact on many applications ranging from statistics and machine learning to computational modeling in physics, chemistry, biology and finance.
We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with P the Markov chain transition matrix and D = \sqrt{P\circ P^T} its discriminant matrix (D=P if P is symmetric), we construct a quantum walk algorithm that for any quantum state |v> and integer t returns a quantum state \epsilon-close to the state D^t|v>/\|D^t|v>. The algorithm uses O(|D^t|v>|^{-1}\sqrt{t\log(\epsilon\|D^t|v>})^{-1}}) expected quantum walk steps and O(\|D^t|v>|^{-1}) expected reflections around |v>. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.
We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with P the Markov chain transition matrix and D=P∘PT its discriminant matrix (D=P if P is symmetric), we construct a quantum walk algorithm that for any quantum state ∣v⟩ and integer t returns a quantum state ε-close to the state Dt∣v⟩/∥Dt∣v⟩∥. The algorithm uses O(∥Dt∣v⟩∥−1tlog(ε∥Dt∣v⟩∥)−1) expected quantum walk steps and O(∥Dt∣v⟩∥−1) expected reflections around ∣v⟩. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.
Preparing quantum samples (QSAMPLES), coherent encodings of stationary distributions of reversible Markov chains, is a fundamental primitive in quantum sampling, particularly for quantum simulated annealing. A central limitation of existing phase-estimation-based frameworks is the ancilla qubit overhead. In this work, we present a new end-to-end framework requiring only one ancilla qubit in the working register. The key technical ingredient is a selective phase compiler circuit using one ancilla qubit, built from a generalized quantum signal processing (GQSP)-based projector onto the 1-eigenspace of the qubitized Szegedy walk. Embedding these selective phase compilers into the fixed-point amplitude amplification (FPAA) procedure and iterating yields a quantum algorithm that, given an initial state, oracle access, lower bounds on the overlaps between adjacent states, and lower bounds on the phase gaps, outputs a QSAMPLE within any desired trace distance and thus total variation distance. The query complexity scales inversely with the square roots of both the minimum overlap and the minimum spectral gap of the Markov chains across the cooling schedule, up to polylogarithmic factors. We also perform simulations to verify how our qubit and query complexity evolve with the trace distance, and how this work compares to the previous framework. These results establish two improvements over the previous framework by Wocjan and Abeyesinghe. First, the working-register ancilla cost is reduced to one. Second, by inserting our GQSP-based selective phase compiler into the FPAA procedure, we improve the QSAMPLE transport overlap dependence from inverse minimum overlap to inverse square-root minimum overlap, relative to their Grover pi-over-three fixed-point method. Finally, as a direct application, we apply the quantum algorithm to prepare a Gibbs QSAMPLE and obtain a rigorous complexity analysis.