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 P has R(P)=O(Q(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(P)=O(Q(P)3l) for any l-uniform hypergraph property 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.

1 aChilds, Andrew, M.1 aWang, Daochen uhttps://arxiv.org/abs/2001.1052001219nas a2200145 4500008004100000245005900041210005800100260001400158520079000172100001800962700001600980700001700996700002301013856003701036 2020 eng d00aQuantum exploration algorithms for multi-armed bandits0 aQuantum exploration algorithms for multiarmed bandits c7/14/20203 aIdentifying 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 O~(∑ni=2Δ−2i−−−−−−−−√) quantum queries, where Δi represents the difference between the mean reward of the best arm and the ith-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).

1 aWang, Daochen1 aYou, Xuchen1 aLi, Tongyang1 aChilds, Andrew, M. uhttps://arxiv.org/abs/2007.0704901555nas a2200169 4500008004100000245005500041210005300096260001400149520105800163100002201221700002301243700001901266700002401285700002101309700001801330856003701348 2020 eng d00aSymmetries, graph properties, and quantum speedups0 aSymmetries graph properties and quantum speedups c6/23/20203 aAaronson 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).

The problem of finding the ground state energy of a Hamiltonian using a quantum computer is currently solved using either the quantum phase estimation (QPE) or variational quantum eigensolver (VQE) algorithms. For precision ε, QPE requires O(1) repetitions of circuits with depth O(1/ε), whereas each expectation estimation subroutine within VQE requires O(1/ε2) samples from circuits with depth O(1). We propose a generalised VQE algorithm that interpolates between these two regimes via a free parameter α∈[0,1] which can exploit quantum coherence over a circuit depth of O(1/εα) to reduce the number of samples to O(1/ε2(1−α)). Along the way, we give a new routine for expectation estimation under limited quantum resources that is of independent interest.

1 aWang, Daochen1 aHiggott, Oscar1 aBrierley, Stephen uhttps://arxiv.org/abs/1802.0017102096nas a2200169 4500008004100000245005300041210005300094260001500147520160200162100002201764700002601786700001801812700001801830700001901848700002201867856003701889 2019 eng d00aEfficient quantum measurement of Pauli operators0 aEfficient quantum measurement of Pauli operators c08/19/20193 aEstimating the expectation value of an observable is a fundamental task in quantum computation. Unfortunately, it is often impossible to obtain such estimates directly, as the computer is restricted to measuring in a fixed computational basis. One common solution splits the observable into a weighted sum of Pauli operators and measures each separately, at the cost of many measurements. An improved version first groups mutually commuting Pauli operators together and then measures all operators within each group simultaneously. The effectiveness of this depends on two factors. First, to enable simultaneous measurement, circuits are required to rotate each group to the computational basis. In our work, we present two efficient circuit constructions that suitably rotate any group of k commuting n-qubit Pauli operators using at most kn−k(k+1)/2 and O(kn/logk) two-qubit gates respectively. Second, metrics that justifiably measure the effectiveness of a grouping are required. In our work, we propose two natural metrics that operate under the assumption that measurements are distributed optimally among groups. Motivated by our new metrics, we introduce SORTED INSERTION, a grouping strategy that is explicitly aware of the weighting of each Pauli operator in the observable. Our methods are numerically illustrated in the context of the Variational Quantum Eigensolver, where the observables in question are molecular Hamiltonians. As measured by our metrics, SORTED INSERTION outperforms four conventional greedy colouring algorithms that seek the minimum number of groups.

1 aCrawford, Ophelia1 avan Straaten, Barnaby1 aWang, Daochen1 aParks, Thomas1 aCampbell, Earl1 aBrierley, Stephen uhttps://arxiv.org/abs/1908.0694200941nas a2200109 4500008004100000245005400041210005400095260001500149520061200164100001800776856003700794 2019 eng d00aSimulating quantum circuits by classical circuits0 aSimulating quantum circuits by classical circuits c04/10/20193 aIn a recent breakthrough, Bravyi, Gosset and König (BGK) [Science, 2018] proved that "simulating" constant depth quantum circuits takes classical circuits Ω(logn) depth. In our paper, we first formalise their notion of simulation, which we call "possibilistic simulation". Then, from well-known results, we deduce that their circuits can be simulated in depth O(log2n). Separately, we construct explicit classical circuits that can simulate any depth-d quantum circuit with Clifford and t T-gates in depth O(d+t). Our classical circuits use {NOT, AND, OR} gates of fan-in ≤2.

1 aWang, Daochen uhttps://arxiv.org/abs/1904.0528201313nas a2200145 4500008004100000245005400041210005400095260001500149490000600164520090100170100001901071700001801090700002201108856003701130 2019 eng d00aVariational Quantum Computation of Excited States0 aVariational Quantum Computation of Excited States c06/28/20190 v33 aThe calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited state energies and their degeneracies. We propose an implementation that requires the same number of qubits as VQE and at most twice the circuit depth. Our method is robust to control errors, is compatible with error-mitigation strategies and can be implemented on near-term quantum compute

1 aHiggott, Oscar1 aWang, Daochen1 aBrierley, Stephen uhttps://arxiv.org/abs/1805.0813801211nas a2200145 4500008004100000245006900041210006900110260001500179490000700194520077300201100001800974700001800992700001801010856003701028 2015 eng d00aDriving Rabi oscillations at the giant dipole resonance in xenon0 aDriving Rabi oscillations at the giant dipole resonance in xenon c11/23/20150 v923 aFree-electron lasers (FELs) produce short and very intense light pulses in the XUV and x-ray regimes. We investigate the possibility to drive Rabi oscillations in xenon with an intense FEL pulse by using the unusually large dipole strength of the giant-dipole resonance (GDR). The GDR decays within less than 30 as due to its position, which is above the 4d ionization threshold. We find that intensities around 1018 W/cm2 are required to induce Rabi oscillations with a period comparable to the lifetime. The pulse duration should not exceed 100 as because xenon will be fully ionized within a few lifetimes. Rabi oscillations reveal themselves also in the photoelectron spectrum in form of Autler-Townes splittings extending over several tens of electronvolt.

1 aPabst, Stefan1 aWang, Daochen1 aSantra, Robin uhttps://arxiv.org/abs/1511.00058