arXiv subjects

Ashley Montanaro

Publications and source records attributed to Ashley Montanaro.

At least 55 records · Page 3Linked to original sources

arXiv2021

Quantum algorithms for learning a hidden graph and beyond

We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model ("OR queries"), the oracle returns whether a given subset of the vertices contains any edges. In the second ("parity queries"), the oracle returns the parity of the number of edges in a subset. In the third model, we are given copies of the graph state corresponding to the graph. We give quantum algorithms that achieve speedups over the best possible classical algorithms in the OR and parity query models, for some families of graphs, and give quantum algorithms in the graph state model whose complexity is similar to the parity query model. For some parameter regimes, the speedups can be exponential in the parity query model. On the other hand, without any promise on the graph, no speedup is possible in the OR query model. A main technique we use is the quantum algorithm for solving the combinatorial group testing problem, for which a query-efficient quantum algorithm was given by Belovs. Here we additionally give a time-efficient quantum algorithm for this problem, based on the algorithm of Ambainis et al.\ for a "gapped" version of the group testing problem. We also give simple time-efficient quantum algorithms based on Fourier sampling and amplitude amplification for learning the exact-half and majority functions, which almost match the optimal complexity of Belovs' algorithms.

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arXiv2020

Strategies for solving the Fermi-Hubbard model on near-term quantum computers

The Fermi-Hubbard model is of fundamental importance in condensed-matter physics, yet is extremely challenging to solve numerically. Finding the ground state of the Hubbard model using variational methods has been predicted to be one of the first applications of near-term quantum computers. Here we carry out a detailed analysis and optimisation of the complexity of variational quantum algorithms for finding the ground state of the Hubbard model, including costs associated with mapping to a real-world hardware platform. The depth complexities we find are substantially lower than previous work. We performed extensive numerical experiments for systems with up to 12 sites. The results suggest that the variational ansätze we used -- an efficient variant of the Hamiltonian Variational ansatz and a novel generalisation thereof -- will be able to find the ground state of the Hubbard model with high fidelity in relatively low quantum circuit depth. Our experiments include the effect of realistic measurements and depolarising noise. If our numerical results on small lattice sizes are representative of the somewhat larger lattices accessible to near-term quantum hardware, they suggest that optimising over quantum circuits with a gate depth less than a thousand could be sufficient to solve instances of the Hubbard model beyond the capacity of classical exact diagonalisation.

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arXiv2020

Quantum vs. classical algorithms for solving the heat equation

Quantum computers are predicted to outperform classical ones for solving partial differential equations, perhaps exponentially. Here we consider a prototypical PDE - the heat equation in a rectangular region - and compare in detail the complexities of ten classical and quantum algorithms for solving it, in the sense of approximately computing the amount of heat in a given region. We find that, for spatial dimension $d \ge 2$, there is an at most quadratic quantum speedup using an approach based on applying amplitude estimation to an accelerated classical random walk. However, an alternative approach based on a quantum algorithm for linear equations is never faster than the best classical algorithms.

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arXiv2020

Compressed variational quantum eigensolver for the Fermi-Hubbard model

The Fermi-Hubbard model is a plausible target to be solved by a quantum computer using the variational quantum eigensolver algorithm. However, problem sizes beyond the reach of classical exact diagonalisation are also beyond the reach of current quantum computing hardware. Here we use a simple method which compresses the first nontrivial subcase of the Hubbard model -- with one spin-up and one spin-down fermion -- enabling larger instances to be addressed using current quantum computing hardware. We implement this method on a superconducting quantum hardware platform for the case of the $2 \times 1$ Hubbard model, including error-mitigation techniques, and show that the ground state is found with relatively high accuracy.

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arXiv2020

Quantum speedups of some general-purpose numerical optimisation algorithms

We give quantum speedups of several general-purpose numerical optimisation methods for minimising a function $f:\mathbb{R}^n \to \mathbb{R}$. First, we show that many techniques for global optimisation under a Lipschitz constraint can be accelerated near-quadratically. Second, we show that backtracking line search, an ingredient in quasi-Newton optimisation algorithms, can be accelerated up to quadratically. Third, we show that a component of the Nelder-Mead algorithm can be accelerated by up to a multiplicative factor of $O(\sqrt{n})$. Fourth, we show that a quantum gradient computation algorithm of Gilyén et al. can be used to approximately compute gradients in the framework of stochastic gradient descent. In each case, our results are based on applying existing quantum algorithms to accelerate specific components of the classical algorithms, rather than developing new quantum techniques.

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arXiv2019

Pretty simple bounds on quantum state discrimination

We show that the quantum measurement known as the pretty good measurement can be used to identify an unknown quantum state picked from any set of $n$ mixed states that have pairwise fidelities upper-bounded by a constant below 1, given $O(\log n)$ copies of the unknown state, with high success probability in the worst case. If the unknown state is promised to be pure, there is an explicit measurement strategy which solves this worst-case quantum state discrimination problem with $\widetilde{O}(\|G\|)$ copies, where $G$ is the Gram matrix of the states.

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arXiv2019

Applying quantum algorithms to constraint satisfaction problems

Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best classical algorithms and taking into account realistic hardware parameters and overheads for fault-tolerance. All known examples of such speedups correspond to problems related to simulation of quantum systems and cryptography. Here we apply general-purpose quantum algorithms for solving constraint satisfaction problems to two families of prototypical NP-complete problems: boolean satisfiability and graph colouring. We consider two quantum approaches: Grover's algorithm and a quantum algorithm for accelerating backtracking algorithms. We compare the performance of optimised versions of these algorithms, when applied to random problem instances, against leading classical algorithms. Even when considering only problem instances that can be solved within one day, we find that there are potentially large quantum speedups available. In the most optimistic parameter regime we consider, this could be a factor of over $10^5$ relative to a classical desktop computer; in the least optimistic regime, the speedup is reduced to a factor of over $10^3$. However, the number of physical qubits used is extremely large, and improved fault-tolerance methods will likely be needed to make these results practical. In particular, the quantum advantage disappears if one includes the cost of the classical processing power required to perform decoding of the surface code using current techniques.

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arXiv2019

Quantum states cannot be transmitted efficiently classically

We show that any classical two-way communication protocol with shared randomness that can approximately simulate the result of applying an arbitrary measurement (held by one party) to a quantum state of $n$ qubits (held by another), up to constant accuracy, must transmit at least $Ω(2^n)$ bits. This lower bound is optimal and matches the complexity of a simple protocol based on discretisation using an $ε$-net. The proof is based on a lower bound on the classical communication complexity of a distributed variant of the Fourier sampling problem. We obtain two optimal quantum-classical separations as easy corollaries. First, a sampling problem which can be solved with one quantum query to the input, but which requires $Ω(N)$ classical queries for an input of size $N$. Second, a nonlocal task which can be solved using $n$ Bell pairs, but for which any approximate classical solution must communicate $Ω(2^n)$ bits.

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arXiv2019

Quantum speedup of branch-and-bound algorithms

Branch-and-bound is a widely used technique for solving combinatorial optimisation problems where one has access to two procedures: a branching procedure that splits a set of potential solutions into subsets, and a cost procedure that determines a lower bound on the cost of any solution in a given subset. Here we describe a quantum algorithm that can accelerate classical branch-and-bound algorithms near-quadratically in a very general setting. We show that the quantum algorithm can find exact ground states for most instances of the Sherrington-Kirkpatrick model in time $O(2^{0.226n})$, which is substantially more efficient than Grover's algorithm.

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arXiv2019

Universal Quantum Hamiltonians

Quantum many-body systems exhibit an extremely diverse range of phases and physical phenomena. Here, we prove that the entire physics of any other quantum many-body system is replicated in certain simple, "universal" spin-lattice models. We first characterise precisely what it means for one quantum many-body system to replicate the entire physics of another. We then show that certain very simple spin-lattice models are universal in this very strong sense. Examples include the Heisenberg and XY models on a 2D square lattice (with non-uniform coupling strengths). We go on to fully classify all two-qubit interactions, determining which are universal and which can only simulate more restricted classes of models. Our results put the practical field of analogue Hamiltonian simulation on a rigorous footing and take a significant step towards justifying why error correction may not be required for this application of quantum information technology.

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arXiv2018

Quantum Computational Supremacy

The field of quantum algorithms aims to find ways to speed up the solution of computational problems by using a quantum computer. A key milestone in this field will be when a universal quantum computer performs a computational task that is beyond the capability of any classical computer, an event known as quantum supremacy. This would be easier to achieve experimentally than full-scale quantum computing, but involves new theoretical challenges. Here we present the leading proposals to achieve quantum supremacy, and discuss how we can reliably compare the power of a classical computer to the power of a quantum computer.

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arXiv2018

A Survey of Quantum Property Testing

The area of property testing tries to design algorithms that can efficiently handle very large amounts of data: given a large object that either has a certain property or is somehow "far" from having that property, a tester should efficiently distinguish between these two cases. In this survey we describe recent results obtained for quantum property testing. This area naturally falls into three parts. First, we may consider quantum testers for properties of classical objects. We survey the main examples known where quantum testers can be much (sometimes exponentially) more efficient than classical testers. Second, we may consider classical testers of quantum objects. This is the situation that arises for instance when one is trying to determine if quantum states or operations do what they are supposed to do, based only on classical input-output behavior. Finally, we may also consider quantum testers for properties of quantum objects, such as states or operations. We survey known bounds on testing various natural properties, such as whether two states are equal, whether a state is separable, whether two operations commute, etc. We also highlight connections to other areas of quantum information theory and mention a number of open questions.

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arXiv2018

Optimal verification of entangled states with local measurements

Consider the task of verifying that a given quantum device, designed to produce a particular entangled state, does indeed produce that state. One natural approach would be to characterise the output state by quantum state tomography; or alternatively to perform some kind of Bell test, tailored to the state of interest. We show here that neither approach is optimal amongst local verification strategies for two qubit states. We find the optimal strategy in this case and show that quadratically fewer total measurements are needed to verify to within a given fidelity than in published results for quantum state tomography, Bell test, or fidelity estimation protocols. We also give efficient verification protocols for any stabilizer state. Additionally, we show that requiring that the strategy be constructed from local, non-adaptive and non-collective measurements only incurs a constant-factor penalty over a strategy without these restrictions.

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arXiv2018

Universal qudit Hamiltonians

A family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits ($d$-level systems): (1) We completely characterise the $k$-qudit interactions which are universal, if augmented with arbitrary 1-local terms. We find that, for all $k \geqslant 2$ and all local dimensions $d \geqslant 2$, almost all such interactions are universal aside from a simple stoquastic class. (2) We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the $SU(d)$ and $SU(2)$ Heisenberg interactions are universal for all local dimensions $d \geqslant 2$ (spin $\geqslant 1/2$), implying that a quantum variant of the Max-$d$-Cut problem is QMA-complete. We also show that for $d=3$ all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. (3) We prove universality of any interaction proportional to the projector onto a pure entangled state.

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arXiv2017

Learning stabilizer states by Bell sampling

We show that measuring pairs of qubits in the Bell basis can be used to obtain a simple quantum algorithm for efficiently identifying an unknown stabilizer state of n qubits. The algorithm uses O(n) copies of the input state and fails with exponentially small probability.

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arXiv2017

Quantum speedup of Monte Carlo methods

Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomised or quantum subroutine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently.

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arXiv2017

The Quantum Complexity of Computing Schatten $p$-norms

We consider the quantum complexity of computing Schatten $p$-norms and related quantities, and find that the problem of estimating these quantities is closely related to the one clean qubit model of computation. We show that the problem of approximating $\text{Tr}\, (|A|^p)$ for a log-local $n$-qubit Hamiltonian $A$ and $p=\text{poly}(n)$, up to a suitable level of accuracy, is contained in DQC1; and that approximating this quantity up to a somewhat higher level of accuracy is DQC1-hard. In some cases the level of accuracy achieved by the quantum algorithm is substantially better than a natural classical algorithm for the problem. The same problem can be solved for arbitrary sparse matrices in BQP. One application of the algorithm is the approximate computation of the energy of a graph.

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arXiv2017

No imminent quantum supremacy by boson sampling

It is predicted that quantum computers will dramatically outperform their conventional counterparts. However, large-scale universal quantum computers are yet to be built. Boson sampling is a rudimentary quantum algorithm tailored to the platform of photons in linear optics, which has sparked interest as a rapid way to demonstrate this quantum supremacy. Photon statistics are governed by intractable matrix functions known as permanents, which suggests that sampling from the distribution obtained by injecting photons into a linear-optical network could be solved more quickly by a photonic experiment than by a classical computer. The contrast between the apparently awesome challenge faced by any classical sampling algorithm and the apparently near-term experimental resources required for a large boson sampling experiment has raised expectations that quantum supremacy by boson sampling is on the horizon. Here we present classical boson sampling algorithms and theoretical analyses of prospects for scaling boson sampling experiments, showing that near-term quantum supremacy via boson sampling is unlikely. While the largest boson sampling experiments reported so far are with 5 photons, our classical algorithm, based on Metropolised independence sampling (MIS), allowed the boson sampling problem to be solved for 30 photons with standard computing hardware. We argue that the impact of experimental photon losses means that demonstrating quantum supremacy by boson sampling would require a step change in technology.

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