We describe algorithms to obtain an approximate classical description of a d-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for ℓq-norm error up to logarithmic factors. As a special case, we show that it takes applications of the unitaries to obtain an ε-ℓ2-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. We characterize the query complexity for obtaining Schatten q-norm estimates of a rank-r mixed state, up to polylogarithmic factors. In particular, we show that a trace-norm (q = 1) estimate can be obtained with queries. This improves (assuming our stronger input model) the ε-dependence over the works of O'Donnell and Wright (STOC 2016) and Haah et al. (IEEE Trans. Inf. Theory, 63.9, 2017), that use a joint measurement on copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to 1 in generic mixed-state tomography algorithms, which can require a large amount of computing resources. We describe sample-optimal algorithms for pure states that are simple and fast to implement. Along the way we show that an ℓ∞-norm estimate of a normalized vector induces a (slightly worse) ℓq-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al. (arXiv:2111.09283) when the measurement operators do not fully overlap. More specifically, we show that for E1,…, Em normalized measurement operators, all expectation values Tr(Ejρ) can be efficiently learned up to error ε with applications of a state-preparation unitary for a purification of ρ. * The full version of the paper can be accessed at https://arxiv.org/abs/2207.08800

The Quantum CONGEST model is a variant of the CONGEST model, where messages consist of O(log(n)) qubits. In this paper, we give a general framework for implementing quantum query algorithms efficiently in a Quantum CONGEST network, using the concept of parallel-query quantum algorithms.

Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorn’s algorithm for matrix scaling and Osborne’s algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time Õ(√{mn}/ε⁴) for scaling or balancing an n × n matrix (given by an oracle) with m non-zero entries to within 𝓁₁-error ε. Their classical analogs use time Õ(m/ε²), and every classical algorithm for scaling or balancing with small constant ε requires Ω(m) queries to the entries of the input matrix. We thus achieve a polynomial speed-up in terms of n, at the expense of a worse polynomial dependence on the obtained 𝓁₁-error ε. Even for constant ε these problems are already non-trivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorn’s and Osborne’s algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorn’s algorithm for entrywise-positive matrices to the 𝓁₁-setting, obtaining an Õ(n^{1.5}/ε³)-time quantum algorithm for ε-𝓁₁-scaling. We also prove a lower bound, showing our quantum algorithm for matrix scaling is essentially optimal for constant ε: every quantum algorithm for matrix scaling that achieves a constant 𝓁₁-error w.r.t. uniform marginals needs Ω(√{mn}) queries.

Following the first paper on quantum algorithms for SDP-solving by Brandão and Svore in 2016, rapid developments has been made on quantum optimization algorithms. Recently Brandão et al. improved the quantum SDP-solver in the so-called quantum state input model, where the input matrices of the SDP are given as purified mixed states. They also gave the first non-trivial application of quantum SDP-solving by obtaining a more efficient algorithm for the problem of shadow tomography (proposed by Aaronson in 2017). In this paper we improve on all previous quantum SDP-solvers. Mainly we construct better Gibbs-samplers for both input models, which directly gives better bounds for SDP-solving. For an SDP with $m$ constraints involving $n\times n$ matrices, our improvements yield an $\widetilde{\mathcal O}\left( \left( \sqrt{m} + \sqrt{n}γ\right)s γ^4\right)$ upper bound on SDP-solving in the sparse matrix input model and an $\widetilde{\mathcal O}\left( \left(\sqrt{m}+B^{2.5}γ^{3.5} \right)Bγ^4 \right)$ upper bound in the quantum state input model. We then apply these results to the problem of shadow tomography to simultaneously improve the best known upper bounds on sample complexity due to Aaronson and complexity due Brandao et al. Furthermore, we apply our quantum SDP-solvers to the problems of quantum state discrimination and E-optimal design. In both cases we beat the classical lower bound in terms of some parameters, at the expense of heavy dependence on some other parameters. Finally we prove two lowers bounds for solving SDPs using quantum algorithms: (1) $\tildeΩ(\sqrt{m}B/\eps)$ in the quantum state input model, and (2) $\tildeΩ(\sqrt{m}α/\eps)$ in the quantum operator input model. These lower bounds show that the $\sqrt{m}$ factor and the polynomial dependence on the parameters $B,α$, and $1/\eps$ are necessary.

Brandao and Svore recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension n of the problem and the number m of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mn when m is approximately n, which is the same as classical.