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Quantum entanglement is an essential feature of many-body systems that impacts both quantum information processing and fundamental physics. The growth of entanglement is a major challenge for classical simulation methods. In this work, we investigate the relationship between quantum entanglement and quantum simulation, showing that product-formula approximations can perform better for entangled systems. We establish a tighter upper bound for algorithmic error in terms of entanglement entropy and develop an adaptive simulation algorithm incorporating measurement gadgets to estimate the algorithmic error. This shows that entanglement is not only an obstacle to classical simulation, but also a feature that can accelerate quantum algorithms., Comment: 31 pages, 6 figures

Geometric locality is an important theoretical and practical factor for quantum low-density parity-check (qLDPC) codes which affects code performance and ease of physical realization. For device architectures restricted to 2D local gates, naively implementing the high-rate codes suitable for low-overhead fault-tolerant quantum computing incurs prohibitive overhead. In this work, we present an error correction protocol built on a bilayer architecture that aims to reduce operational overheads when restricted to 2D local gates by measuring some generators less frequently than others. We investigate the family of bivariate bicycle qLDPC codes and show that they are well suited for a parallel syndrome measurement scheme using fast routing with local operations and classical communication (LOCC). Through circuit-level simulations, we find that in some parameter regimes bivariate bicycle codes implemented with this protocol have logical error rates comparable to the surface code while using fewer physical qubits., Comment: 16 pages, 10 figures

In this work we propose an approach for implementing time-evolution of a quantum system using product formulas. The quantum algorithms we develop have provably better scaling (in terms of gate complexity and circuit depth) than a naive application of well-known Trotter formulas, for systems where the evolution is determined by a Hamiltonian with different energy scales (i.e., one part is "large" and another part is "small"). Our algorithms generate a decomposition of the evolution operator into a product of simple unitaries that are directly implementable on a quantum computer. Although the theoretical scaling is suboptimal compared with state-of-the-art algorithms (e.g., quantum signal processing), the performance of the algorithms we propose is highly competitive in practice. We illustrate this via extensive numerical simulations for several models. For instance, in the strong-field regime of the 1D transverse-field Ising model, our algorithms achieve an improvement of one order of magnitude in both the system size and evolution time that can be simulated with a fixed budget of 1000 arbitrary 2-qubit gates, compared with standard Trotter formulas., Comment: 39 pages, 14 figures, 4 tables

Existing schemes for demonstrating quantum computational advantage are subject to various practical restrictions, including the hardness of verification and challenges in experimental implementation. Meanwhile, analog quantum simulators have been realized in many experiments to study novel physics. In this work, we propose a quantum advantage protocol based on single-step Feynman-Kitaev verification of an analog quantum simulation, in which the verifier need only run an $O(\lambda^2)$-time classical computation, and the prover need only prepare $O(1)$ samples of a history state and perform $O(\lambda^2)$ single-qubit measurements, for a security parameter $\lambda$. We also propose a near-term feasible strategy for honest provers and discuss potential experimental realizations., Comment: 20 pages, 6 figures

Quantum computers offer the potential to simulate nuclear processes that are classically intractable. With the goal of understanding the necessary quantum resources, we employ state-of-the-art Hamiltonian-simulation methods, and conduct a thorough algorithmic analysis, to estimate the qubit and gate costs to simulate low-energy effective field theories (EFTs) of nuclear physics. In particular, within the framework of nuclear lattice EFT, we obtain simulation costs for the leading-order pionless and pionful EFTs. We consider both static pions represented by a one-pion-exchange potential between the nucleons, and dynamical pions represented by relativistic bosonic fields coupled to non-relativistic nucleons. We examine the resource costs for the tasks of time evolution and energy estimation for physically relevant scales. We account for model errors associated with truncating either long-range interactions in the one-pion-exchange EFT or the pionic Hilbert space in the dynamical-pion EFT, and for algorithmic errors associated with product-formula approximations and quantum phase estimation. Our results show that the pionless EFT is the least costly to simulate and the dynamical-pion theory is the costliest. We demonstrate how symmetries of the low-energy nuclear Hamiltonians can be utilized to obtain tighter error bounds on the simulation algorithm. By retaining the locality of nucleonic interactions when mapped to qubits, we achieve reduced circuit depth and substantial parallelization. We further develop new methods to bound the algorithmic error for classes of fermionic Hamiltonians that preserve the number of fermions, and demonstrate that reasonably tight Trotter error bounds can be achieved by explicitly computing nested commutators of Hamiltonian terms. This work highlights the importance of combining physics insights and algorithmic advancement in reducing quantum-simulation costs., Comment: 51 pages, 55 pages appendix, 22 figures

We introduce a family of identities that express general linear non-unitary evolution operators as a linear combination of unitary evolution operators, each solving a Hamiltonian simulation problem. This formulation can exponentially enhance the accuracy of the recently introduced linear combination of Hamiltonian simulation (LCHS) method [An, Liu, and Lin, Physical Review Letters, 2023]. For the first time, this approach enables quantum algorithms to solve linear differential equations with both optimal state preparation cost and near-optimal scaling in matrix queries on all parameters.

Quantum state purification is the task of recovering a nearly pure copy of an unknown pure quantum state using multiple noisy copies of the state. This basic task has applications to quantum communication over noisy channels and quantum computation with imperfect devices, but has only been studied previously for the case of qubits. We derive an efficient purification procedure based on the swap test for qudits of any dimension, starting with any initial error parameter. Treating the initial error parameter and the dimension as constants, we show that our procedure has sample complexity asymptotically optimal in the final error parameter. Our protocol has a simple recursive structure that can be applied when the states are provided one at a time in a streaming fashion, requiring only a small quantum memory to implement., Comment: 34 pages, substantially improved and extended results from version 1

The so-called welded tree problem provides an example of a black-box problem that can be solved exponentially faster by a quantum walk than by any classical algorithm. Given the name of a special ENTRANCE vertex, a quantum walk can find another distinguished EXIT vertex using polynomially many queries, though without finding any particular path from ENTRANCE to EXIT. It has been an open problem for twenty years whether there is an efficient quantum algorithm for finding such a path, or if the path-finding problem is hard even for quantum computers. We show that a natural class of efficient quantum algorithms provably cannot find a path from ENTRANCE to EXIT. Specifically, we consider algorithms that, within each branch of their superposition, always store a set of vertex labels that form a connected subgraph including the ENTRANCE, and that only provide these vertex labels as inputs to the oracle. While this does not rule out the possibility of a quantum algorithm that efficiently finds a path, it is unclear how an algorithm could benefit by deviating from this behavior. Our no-go result suggests that, for some problems, quantum algorithms must necessarily forget the path they take to reach a solution in order to outperform classical computation., Comment: 49 pages, 9 figures

Given a convex function $f\colon\mathbb{R}^{d}\to\mathbb{R}$, the problem of sampling from a distribution $\propto e^{-f(x)}$ is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc. In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants $\int_{\mathbb{R}^d}e^{-f(x)}\mathrm{d} x$. First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number $\kappa$ and dimension $d$) of analogous classical algorithms that use gradient (first-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries. For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error $\epsilon$. Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity $\widetilde{O}(\kappa^{1/2}d)$ and $\widetilde{O}(\kappa^{1/2}d^{3/2}/\epsilon)$ for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in $\kappa,d,\epsilon$ over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/\epsilon^{1-o(1)}$ quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in $\epsilon$., Comment: To appear in the proceedings of NeurIPS 2022

The divide-and-conquer framework, used extensively in classical algorithm design, recursively breaks a problem of size $n$ into smaller subproblems (say, $a$ copies of size $n/b$ each), along with some auxiliary work of cost $C^{\textrm{aux}}(n)$, to give a recurrence relation $$C(n) \leq a \, C(n/b) + C^{\textrm{aux}}(n)$$ for the classical complexity $C(n)$. We describe a quantum divide-and-conquer framework that, in certain cases, yields an analogous recurrence relation $$C_Q(n) \leq \sqrt{a} \, C_Q(n/b) + O(C^{\textrm{aux}}_Q(n))$$ that characterizes the quantum query complexity. We apply this framework to obtain near-optimal quantum query complexities for various string problems, such as (i) recognizing regular languages; (ii) decision versions of String Rotation and String Suffix; and natural parameterized versions of (iii) Longest Increasing Subsequence and (iv) Longest Common Subsequence., Comment: 24 pages, 8 figures

The Swap gate is a ubiquitous tool for moving information on quantum hardware, yet it can be considered a classical operation because it does not entangle product states. Genuinely quantum operations could outperform Swap for the task of permuting qubits within an architecture, which we call routing. We consider quantum routing in two models: (1) allowing arbitrary two-qubit unitaries, or (2) allowing Hamiltonians with norm-bounded interactions. We lower bound the circuit depth or time of quantum routing in terms of spectral properties of graphs representing the architecture interaction constraints, and give a generalized upper bound for all simple connected $n$-vertex graphs. In particular, we give conditions for a superpolynomial classical-quantum routing separation, which exclude graphs with a small spectral gap and graphs of bounded degree. Finally, we provide examples of a quadratic separation between gate-based and Hamiltonian routing models with a constant number of local ancillas per qubit and of an $\Omega(n)$ speedup if we also allow fast local interactions., Comment: 45 pages, 7 figures

We study the problem of implementing arbitrary permutations of qubits under interaction constraints in quantum systems that allow for arbitrarily fast local operations and classical communication (LOCC). In particular, we show examples of speedups over swap-based and more general unitary routing methods by distributing entanglement and using LOCC to perform quantum teleportation. We further describe an example of an interaction graph for which teleportation gives a logarithmic speedup in the worst-case routing time over swap-based routing. We also study limits on the speedup afforded by quantum teleportation - showing an $O(\sqrt{N \log N})$ upper bound on the separation in routing time for any interaction graph - and give tighter bounds for some common classes of graphs., Comment: 12 pages, 7 figures

Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finite-dimensional systems, less is known about quantum algorithms for real-space dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a $d$-dimensional Schr\"{o}dinger equation with $\eta$ particles can be simulated with gate complexity $\tilde{O}\bigl(\eta d F \text{poly}(\log(g'/\epsilon))\bigr)$, where $\epsilon$ is the discretization error, $g'$ controls the higher-order derivatives of the wave function, and $F$ measures the time-integrated strength of the potential. Compared to the best previous results, this exponentially improves the dependence on $\epsilon$ and $g'$ from $\text{poly}(g'/\epsilon)$ to $\text{poly}(\log(g'/\epsilon))$ and polynomially improves the dependence on $T$ and $d$, while maintaining best known performance with respect to $\eta$. For the case of Coulomb interactions, we give an algorithm using $\eta^{3}(d+\eta)T\text{poly}(\log(\eta dTg'/(\Delta\epsilon)))/\Delta$ one- and two-qubit gates, and another using $\eta^{3}(4d)^{d/2}T\text{poly}(\log(\eta dTg'/(\Delta\epsilon)))/\Delta$ one- and two-qubit gates and QRAM operations, where $T$ is the evolution time and the parameter $\Delta$ regulates the unbounded Coulomb interaction. We give applications to several computational problems, including faster real-space simulation of quantum chemistry, rigorous analysis of discretization error for simulation of a uniform electron gas, and a quadratic improvement to a quantum algorithm for escaping saddle points in nonconvex optimization.

Quantum walks provide a framework for understanding and designing quantum algorithms that is both intuitive and universal. To leverage the computational power of these walks, it is important to be able to programmably modify the graph a walker traverses while maintaining coherence. Here, we do this by combining the fast, programmable control provided by optical tweezer arrays with the scalable, homogeneous environment of an optical lattice. Using this new combination of tools we study continuous-time quantum walks of single atoms on a 2D square lattice, and perform proof-of-principle demonstrations of spatial search using these walks. When scaled to more particles, the capabilities demonstrated here can be extended to study a variety of problems in quantum information science and quantum simulation, including the deterministic assembly of ground and excited states in Hubbard models with tunable interactions, and performing versions of spatial search in a larger graph with increased connectivity, where search by quantum walk can be more effective., Comment: 9 pages, 3 figures (main text); 13 pages, 8 figures (supplemental materials)

We construct product formulas for exponentials of commutators and explore their applications. First, we directly construct a third-order product formula with six exponentials by solving polynomial equations obtained using the operator differential method. We then derive higher-order product formulas recursively from the third-order formula. We improve over previous recursive constructions, reducing the number of gates required to achieve the same accuracy. In addition, we demonstrate that the constituent linear terms in the commutator can be included at no extra cost. As an application, we show how to use the product formulas in a digital protocol for counterdiabatic driving, which increases the fidelity for quantum state preparation. We also discuss applications to quantum simulation of one-dimensional fermion chains with nearest- and next-nearest-neighbor hopping terms, and two-dimensional fractional quantum Hall phases., Comment: 18 pages, 11 figures

The algorithmic error of digital quantum simulations is usually explored in terms of the spectral norm distance between the actual and ideal evolution operators. In practice, this worst-case error analysis may be unnecessarily pessimistic. To address this, we develop a theory of average-case performance of Hamiltonian simulation with random initial states. We relate the average-case error to the Frobenius norm of the multiplicative error and give upper bounds for the product formula (PF) and truncated Taylor series methods. As applications, we estimate average-case error for digital Hamiltonian simulation of general lattice Hamiltonians and $k$-local Hamiltonians. In particular, for the nearest-neighbor Heisenberg chain with $n$ spins, the error is quadratically reduced from $\mathcal O(n)$ in the worst case to $\mathcal O(\sqrt{n})$ on average for both the PF method and the Taylor series method. Numerical evidence suggests that this theory accurately characterizes the average error for concrete models. We also apply our results to error analysis in the simulation of quantum scrambling.

We present methods for implementing arbitrary permutations of qubits under interaction constraints. Our protocols make use of previous methods for rapidly reversing the order of qubits along a path. Given nearest-neighbor interactions on a path of length $n$, we show that there exists a constant $\epsilon \approx 0.034$ such that the quantum routing time is at most $(1-\epsilon)n$, whereas any swap-based protocol needs at least time $n-1$. This represents the first known quantum advantage over swap-based routing methods and also gives improved quantum routing times for realistic architectures such as grids. Furthermore, we show that our algorithm approaches a quantum routing time of $2n/3$ in expectation for uniformly random permutations, whereas swap-based protocols require time $n$ asymptotically. Additionally, we consider sparse permutations that route $k \le n$ qubits and give algorithms with quantum routing time at most $n/3 + O(k^2)$ on paths and at most $2r/3 + O(k^2)$ on general graphs with radius $r$., Comment: 26 pages, 10 figures. Updated version forthcoming in Quantum

We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation., Comment: 19 pages. Added discussion of related prior work

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic $n$-dimensional ordinary differential equations. Assuming $R < 1$, where $R$ is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity $T^2 q~\mathrm{poly}(\log T, \log n, \log 1/\epsilon)/\epsilon$, where $T$ is the evolution time, $\epsilon$ is the allowed error, and $q$ measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in $T$. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a novel convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for $R \ge \sqrt{2}$. Finally, we discuss potential applications, showing that the $R < 1$ condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of $R$., Comment: 36 pages, 1 figure. Published in PNAS

Identifying the best arm of a multi-armed bandit is a central problem in bandit optimization. We study a quantum computational version of this problem with coherent oracle access to states encoding the reward probabilities of each arm as quantum amplitudes. Specifically, we show that we can find the best arm with fixed confidence using $\tilde{O}\bigl(\sqrt{\sum_{i=2}^n\Delta^{\smash{-2}}_i}\bigr)$ quantum queries, where $\Delta_{i}$ represents the difference between the mean reward of the best arm and the $i^\text{th}$-best arm. This algorithm, based on variable-time amplitude amplification and estimation, gives a quadratic speedup compared to the best possible classical result. We also prove a matching quantum lower bound (up to poly-logarithmic factors)., Comment: 18 pages, 1 figure. To appear in the Thirty-Fifth AAAI Conference on Artificial Intelligence (AAAI 2021)

The standard circuit model for quantum computation presumes the ability to directly perform gates between arbitrary pairs of qubits, which is unlikely to be practical for large-scale experiments. Power-law interactions with strength decaying as $1/r^\alpha$ in the distance $r$ provide an experimentally realizable resource for information processing, whilst still retaining long-range connectivity. We leverage the power of these interactions to implement a fast quantum fanout gate with an arbitrary number of targets. Our implementation allows the quantum Fourier transform (QFT) and Shor's algorithm to be performed on a $D$-dimensional lattice in time logarithmic in the number of qubits for interactions with $\alpha \le D$. As a corollary, we show that power-law systems with $\alpha \le D$ are difficult to simulate classically even for short times, under a standard assumption that factoring is classically intractable. Complementarily, we develop a new technique to give a general lower bound, linear in the size of the system, on the time required to implement the QFT and the fanout gate in systems that are constrained by a linear light cone. This allows us to prove an asymptotically tighter lower bound for long-range systems than is possible with previously available techniques., Comment: 6 pages, 1 figure

Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups. In contrast, in the adjacency list model for bounded-degree graphs (where graph symmetry is manifested differently), we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013)., Comment: 46 pages. Subsumes arXiv:2001.09642 and arXiv:2001.10520; adds a characterization of permutation groups with speedup and an exponential speedup for adjacency-list graph property testing

We propose a time-independent Hamiltonian protocol for the reversal of qubit ordering in a chain of $N$ spins. Our protocol has an easily implementable nearest-neighbor, transverse-field Ising model Hamiltonian with time-independent, non-uniform couplings. Under appropriate normalization, we implement this state reversal three times faster than a naive approach using SWAP gates, in time comparable to a protocol of Raussendorf [Phys. Rev. A 72, 052301 (2005)] that requires dynamical control. We also prove lower bounds on state reversal by using results on the entanglement capacity of Hamiltonians and show that we are within a factor $1.502(1+1/N)$ of the shortest time possible. Our lower bound holds for all nearest-neighbor qubit protocols with arbitrary finite ancilla spaces and local operations and classical communication. Finally, we extend our protocol to an infinite family of nearest-neighbor, time-independent Hamiltonian protocols for state reversal. This includes chains with nearly uniform coupling that may be especially feasible for experimental implementation., Comment: 7 pages, 2 figures

Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity $\mathrm{poly}(1/\epsilon)$, where $\epsilon$ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be $\mathrm{poly}(d, \log(1/\epsilon))$, where $d$ is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations., Comment: 40 pages. Corrected the dependence on the dimension of the finite difference method

We study how efficiently a $k$-element set $S\subseteq[n]$ can be learned from a uniform superposition $|S\rangle$ of its elements. One can think of $|S\rangle=\sum_{i\in S}|i\rangle/\sqrt{|S|}$ as the quantum version of a uniformly random sample over $S$, as in the classical analysis of the ``coupon collector problem.'' We show that if $k$ is close to $n$, then we can learn $S$ using asymptotically fewer quantum samples than random samples. In particular, if there are $n-k=O(1)$ missing elements then $O(k)$ copies of $|S\rangle$ suffice, in contrast to the $\Theta(k\log k)$ random samples needed by a classical coupon collector. On the other hand, if $n-k=\Omega(k)$, then $\Omega(k\log k)$ quantum samples are~necessary. More generally, we give tight bounds on the number of quantum samples needed for every $k$ and $n$, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through $|S\rangle$. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case., Comment: 17 pages LaTeX

Quantum computers can sometimes exponentially outperform classical ones, but only for problems with sufficient structure. While it is well known that query problems with full permutation symmetry can have at most polynomial quantum speedup -- even for partial functions -- it is unclear how far this condition must be relaxed to enable exponential speedup. In particular, it is natural to ask whether exponential speedup is possible for (partial) graph properties, in which the input describes a graph and the output can only depend on its isomorphism class. We show that the answer to this question depends strongly on the input model. In the adjacency matrix model, we prove that the bounded-error randomized query complexity $R$ of any graph property $\mathcal{P}$ has $R(\mathcal{P}) = O(Q(\mathcal{P})^{6})$, where $Q$ is the bounded-error quantum query complexity. This negatively resolves an open question of Montanaro and de Wolf in the adjacency matrix model. More generally, we prove $R(\mathcal{P}) = O(Q(\mathcal{P})^{3l})$ for any $l$-uniform hypergraph property $\mathcal{P}$ in the adjacency matrix model. In direct contrast, in the adjacency list model for bounded-degree graphs, we exhibit a promise problem that shows an exponential separation between the randomized and quantum query complexities., Comment: 11 pages

Quantum computers can efficiently simulate the dynamics of quantum systems. In this paper, we study the cost of digitally simulating the dynamics of several physically relevant systems using the first-order product formula algorithm. We show that the errors from different Trotterization steps in the algorithm can interfere destructively, yielding a much smaller error than previously estimated. In particular, we prove that the total error in simulating a nearest-neighbor interacting system of $n$ sites for time $t$ using the first-order product formula with $r$ time slices is $O({nt}/{r}+{nt^3}/{r^2})$ when $nt^2/r$ is less than a small constant. Given an error tolerance $\epsilon$, the error bound yields an estimate of $\max\{O({n^2t}/{\epsilon}),O({n^2 t^{3/2}}/{\epsilon^{1/2}})\}$ for the total gate count of the simulation. The estimate is tighter than previous bounds and matches the empirical performance observed in Childs et al. [PNAS 115, 9456-9461 (2018)]. We also provide numerical evidence for potential improvements and conjecture an even tighter estimate for the gate count., Comment: 9 pages, 2 figures

The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure, $k$-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a byproduct. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of $5$, and is close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor., Comment: 82 pages, 5 figures. Enhanced version of the article published in Physical Review X at http://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.011020

In a recent breakthrough, Mahadev constructed an interactive protocol that enables a purely classical party to delegate any quantum computation to an untrusted quantum prover. In this work, we show that this same task can in fact be performed non-interactively and in zero-knowledge. Our protocols result from a sequence of significant improvements to the original four-message protocol of Mahadev. We begin by making the first message instance-independent and moving it to an offline setup phase. We then establish a parallel repetition theorem for the resulting three-message protocol, with an asymptotically optimal rate. This, in turn, enables an application of the Fiat-Shamir heuristic, eliminating the second message and giving a non-interactive protocol. Finally, we employ classical non-interactive zero-knowledge (NIZK) arguments and classical fully homomorphic encryption (FHE) to give a zero-knowledge variant of this construction. This yields the first purely classical NIZK argument system for QMA, a quantum analogue of NP. We establish the security of our protocols under standard assumptions in quantum-secure cryptography. Specifically, our protocols are secure in the Quantum Random Oracle Model, under the assumption that Learning with Errors is quantumly hard. The NIZK construction also requires circuit-private FHE., Comment: 37 pages

Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an $n$-dimensional convex body within multiplicative error $\epsilon$ using $\tilde{O}(n^{3}+n^{2.5}/\epsilon)$ queries to a membership oracle and $\tilde{O}(n^{5}+n^{4.5}/\epsilon)$ additional arithmetic operations. For comparison, the best known classical algorithm uses $\tilde{O}(n^{4}+n^{3}/\epsilon^{2})$ queries and $\tilde{O}(n^{6}+n^{5}/\epsilon^{2})$ additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of "Chebyshev cooling", where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requires $\Omega(\sqrt n+1/\epsilon)$ quantum membership queries, which rules out the possibility of exponential quantum speedup in $n$ and shows optimality of our algorithm in $1/\epsilon$ up to poly-logarithmic factors., Comment: 61 pages, 8 figures. v2: Quantum query complexity improved to $\tilde{O}(n^{3}+n^{2.5}/\epsilon)$ and number of additional arithmetic operations improved to $\tilde{O}(n^{5}+n^{4.5}/\epsilon)$. v3: Improved Section 4.3.3 on nondestructive mean estimation and Section 6 on quantum lower bounds; various minor changes

The difficulty of simulating quantum dynamics depends on the norm of the Hamiltonian. When the Hamiltonian varies with time, the simulation complexity should only depend on this quantity instantaneously. We develop quantum simulation algorithms that exploit this intuition. For sparse Hamiltonian simulation, the gate complexity scales with the $L^1$ norm $\int_{0}^{t}\mathrm{d}\tau\left\lVert H(\tau)\right\lVert_{\max}$, whereas the best previous results scale with $t\max_{\tau\in[0,t]}\left\lVert H(\tau)\right\lVert_{\max}$. We also show analogous results for Hamiltonians that are linear combinations of unitaries. Our approaches thus provide an improvement over previous simulation algorithms that can be substantial when the Hamiltonian varies significantly. We introduce two new techniques: a classical sampler of time-dependent Hamiltonians and a rescaling principle for the Schr\"{o}dinger equation. The rescaled Dyson-series algorithm is nearly optimal with respect to all parameters of interest, whereas the sampling-based approach is easier to realize for near-term simulation. These algorithms could potentially be applied to semi-classical simulations of scattering processes in quantum chemistry., Comment: 40 pages, 1 figure

Strongly long-range interacting quantum systems---those with interactions decaying as a power-law $1/r^{\alpha}$ in the distance $r$ on a $D$-dimensional lattice for $\alpha\le D$---have received significant interest in recent years. They are present in leading experimental platforms for quantum computation and simulation, as well as in theoretical models of quantum information scrambling and fast entanglement creation. Since no notion of locality is expected in such systems, a general understanding of their dynamics is lacking. As a first step towards rectifying this problem, we prove two new Lieb-Robinson-type bounds that constrain the time for signaling and scrambling in strongly long-range interacting systems, for which no tight bounds were previously known. Our first bound applies to systems mappable to free-particle Hamiltonians with long-range hopping, and is saturable for $\alpha\le D/2$. Our second bound pertains to generic long-range interacting spin Hamiltonians, and leads to a tight lower bound for the signaling time to extensive subsets of the system for all $\alpha < D$. This result also lower-bounds the scrambling time, and suggests a path towards achieving a tight scrambling bound that can prove the long-standing fast scrambling conjecture., Comment: $5+\epsilon + 5$ pages, $1+2$ figures

Quantum computer architectures impose restrictions on qubit interactions. We propose efficient circuit transformations that modify a given quantum circuit to fit an architecture, allowing for any initial and final mapping of circuit qubits to architecture qubits. To achieve this, we first consider the qubit movement subproblem and use the routing via matchings framework to prove tighter bounds on parallel routing. In practice, we only need to perform partial permutations, so we generalize routing via matchings to that setting. We give new routing procedures for common architecture graphs and for the generalized hierarchical product of graphs, which produces subgraphs of the Cartesian product. Secondly, for serial routing, we consider the token swapping framework and extend a 4-approximation algorithm for general graphs to support partial permutations. We apply these routing procedures to give several circuit transformations, using various heuristic qubit placement subroutines. We implement these transformations in software and compare their performance for large quantum circuits on grid and modular architectures, identifying strategies that work well in practice., Comment: 29 pages, 1 figure. LuaTeX paper source available at https://gitlab.umiacs.umd.edu/eschoute/arct-paper ; updated with changes from published version, tightened lower bound for modular architectures

Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a $d$-dimensional system of linear equations or linear differential equations with complexity $\mathrm{poly}(\log d)$. While several of these algorithms approximate the solution to within $\epsilon$ with complexity $\mathrm{poly}(\log(1/\epsilon))$, no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity $\mathrm{poly}(\log d, \log(1/\epsilon))$., Comment: 29 pages

We consider simulating an $n$-qubit Hamiltonian with nearest-neighbor interactions evolving for time $t$ on a quantum computer. We show that this simulation has gate complexity $(nt)^{1+o(1)}$ using product formulas, a straightforward approach that has been demonstrated by several experimental groups. While it is reasonable to expect this complexity---in particular, this was claimed without rigorous justification by Jordan, Lee, and Preskill---we are not aware of a straightforward proof. Our approach is based on an analysis of the local error structure of product formulas, as introduced by Descombes and Thalhammer and further simplified here. We prove error bounds for canonical product formulas, which include well-known constructions such as the Lie-Trotter-Suzuki formulas. We also develop a local error representation for time-dependent Hamiltonian simulation, and we discuss generalizations to periodic boundary conditions, constant-range interactions, and higher dimensions. Combined with a previous lower bound, our result implies that product formulas can simulate lattice Hamiltonians with nearly optimal gate complexity., Comment: 24 pages, 3 figures

While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function over an $n$-dimensional convex body using $\tilde{O}(n)$ queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the best-known classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires $\tilde{\Omega}(\sqrt n)$ evaluation queries and $\Omega(\sqrt{n})$ membership queries., Comment: 44 pages, 2 figures. Similar results were independently obtained by Joran van Apeldoorn, Andras Gilyen, Sander Gribling, and Ronald de Wolf

The propagation of information in non-relativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We derive a new Lieb-Robinson bound for systems with interactions that decay with distance $r$ as a power law, $1/r^\alpha$. The bound implies an effective light cone tighter than all previous bounds. Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al., FOCS'18. To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish. This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation. In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions. In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when $\alpha>3D$ (where $D$ is the number of dimensions)., Comment: 18 pages, 10 figures

Product formulas can be used to simulate Hamiltonian dynamics on a quantum computer by approximating the exponential of a sum of operators by a product of exponentials of the individual summands. This approach is both straightforward and surprisingly efficient. We show that by simply randomizing how the summands are ordered, one can prove stronger bounds on the quality of approximation for product formulas of any given order, and thereby give more efficient simulations. Indeed, we show that these bounds can be asymptotically better than previous bounds that exploit commutation between the summands, despite using much less information about the structure of the Hamiltonian. Numerical evidence suggests that the randomized approach has better empirical performance as well., Comment: 20 pages, 2 figures

With quantum computers of significant size now on the horizon, we should understand how to best exploit their initially limited abilities. To this end, we aim to identify a practical problem that is beyond the reach of current classical computers, but that requires the fewest resources for a quantum computer. We consider quantum simulation of spin systems, which could be applied to understand condensed matter phenomena. We synthesize explicit circuits for three leading quantum simulation algorithms, employing diverse techniques to tighten error bounds and optimize circuit implementations. Quantum signal processing appears to be preferred among algorithms with rigorous performance guarantees, whereas higher-order product formulas prevail if empirical error estimates suffice. Our circuits are orders of magnitude smaller than those for the simplest classically-infeasible instances of factoring and quantum chemistry., Comment: 63 pages, 13 figures, 2 tables

We develop and implement automated methods for optimizing quantum circuits of the size and type expected in quantum computations that outperform classical computers. We show how to handle continuous gate parameters and report a collection of fast algorithms capable of optimizing large-scale quantum circuits. For the suite of benchmarks considered, we obtain substantial reductions in gate counts. In particular, we provide better optimization in significantly less time than previous approaches, while making minimal structural changes so as to preserve the basic layout of the underlying quantum algorithms. Our results help bridge the gap between the computations that can be run on existing hardware and those that are expected to outperform classical computers., Comment: 21 pages

How many quantum queries are required to determine the coefficients of a degree-$d$ polynomial in $n$ variables? We present and analyze quantum algorithms for this multivariate polynomial interpolation problem over the fields $\mathbb{F}_q$, $\mathbb{R}$, and $\mathbb{C}$. We show that $k_{\mathbb{C}}$ and $2k_{\mathbb{C}}$ queries suffice to achieve probability $1$ for $\mathbb{C}$ and $\mathbb{R}$, respectively, where $k_{\mathbb{C}}=\smash{\lceil\frac{1}{n+1}{n+d\choose d}\rceil}$ except for $d=2$ and four other special cases. For $\mathbb{F}_q$, we show that $\smash{\lceil\frac{d}{n+d}{n+d\choose d}\rceil}$ queries suffice to achieve probability approaching $1$ for large field order $q$. The classical query complexity of this problem is $\smash{n+d\choose d}$, so our result provides a speedup by a factor of $n+1$, $\frac{n+1}{2}$, and $\frac{n+d}{d}$ for $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{F}_q$, respectively. Thus we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of $2$. For the case of $\mathbb{F}_q$, we conjecture that $2k_{\mathbb{C}}$ queries also suffice to achieve probability approaching $1$ for large field order $q$, although we leave this as an open problem., Comment: 15 pages, 0 figures. Comments are welcome

We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability., Comment: 20 pages, no figure; v2: minor revision

Quantum algorithms for simulating Hamiltonian dynamics have been extensively developed, but there has been much less work on quantum algorithms for simulating the dynamics of open quantum systems. We give the first efficient quantum algorithms for simulating Markovian quantum dynamics generated by Lindbladians that are not necessarily local. We introduce two approaches to simulating sparse Lindbladians. First, we show how to simulate Lindbladians that act within small invariant subspaces using a quantum algorithm to implement sparse Stinespring isometries. Second, we develop a method for simulating sparse Lindblad operators by concatenating a sequence of short-time evolutions. We also show limitations on Lindbladian simulation by proving a no-fast-forwarding theorem for simulating sparse Lindbladians in black-box models., Comment: 48 pages, 6 figures. v2: A complete proof of Proposition 2 is added

Harrow, Hassidim, and Lloyd showed that for a suitably specified $N \times N$ matrix $A$ and $N$-dimensional vector $\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations $A\vec{x}=\vec{b}$. If $A$ is sparse and well-conditioned, their algorithm runs in time $\mathrm{poly}(\log N, 1/\epsilon)$, where $\epsilon$ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in $\log(1/\epsilon)$, exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on $\epsilon$ is prohibitive., Comment: v1: 28 pages; v2: 31 pages, minor change to title, various minor changes and clarifications in response to referee comments

We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over GF(q). A lower bound shown independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2+1/2 quantum queries are needed to solve this problem with bounded error, whereas an algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We show that the lower bound is achievable: d/2+1/2 quantum queries suffice to determine the polynomial with bounded error. Furthermore, we show that d/2+1 queries suffice to achieve probability approaching 1 for large q. These upper bounds improve results of Boneh and Zhandry on the insecurity of cryptographic protocols against quantum attacks. We also show that our algorithm's success probability as a function of the number of queries is precisely optimal. Furthermore, the algorithm can be implemented with gate complexity poly(log q) with negligible decrease in the success probability. We end with a conjecture about the quantum query complexity of multivariate polynomial interpolation., Comment: 17 pages, minor improvements, added conjecture about multivariate interpolation

Nonlinear variants of quantum mechanics can solve tasks that are impossible in standard quantum theory, such as perfectly distinguishing nonorthogonal states. Here we derive the optimal protocol for distinguishing two states of a qubit using the Gross-Pitaevskii equation, a model of nonlinear quantum mechanics that arises as an effective description of Bose-Einstein condensates. Using this protocol, we present an algorithm for unstructured search in the Gross-Pitaevskii model, obtaining an exponential improvement over a previous algorithm of Meyer and Wong. This result establishes a limitation on the effectiveness of the Gross-Pitaevskii approximation. More generally, we demonstrate similar behavior under a family of related nonlinearities, giving evidence that the ability to quickly discriminate nonorthogonal states and thereby solve unstructured search is a generic feature of nonlinear quantum mechanics., Comment: 14 pages, 4 figures; published version, with minor corrections and an improved discussion of the validity of the Gross-Pitaevskii approximation

We prove that approximating the ground energy of the antiferromagnetic XY model on a simple graph at fixed magnetization (given as part of the instance specification) is QMA-complete. To show this, we strengthen a previous result by establishing QMA-completeness for approximating the ground energy of the Bose-Hubbard model on simple graphs. Using a connection between the XY and Bose-Hubbard models that we exploited in previous work, this establishes QMA-completeness of the XY model.