We present a protocol for fault-tolerantly implementing the logical quantum random access memory (QRAM) operation, given access to a specialized, noisy QRAM device. For coherently accessing classical memories of size $2^{n}$, our protocol consumes only poly $(n)$ fault-tolerant quantum resources (logical gates, logical qubits, quantum error correction cycles, etc.), avoiding the need to perform active error correction on all $\Omega\left(2^{n}\right)$ components of the QRAM device. This is the first rigorous conceptual demonstration that a specialized, noisy QRAM device could be useful for implementing a fault-tolerant quantum algorithm. In fact, the fidelity of the device can be as low as $1 / \operatorname{poly}(n)$. The protocol queries the noisy QRAM device $\operatorname{poly}(n)$ times to prepare a sequence of n-qubit QRAM resource states, which are moved to a general-purpose poly $(n)$ size processor to be encoded into a QEC code, distilled, and faulttolerantly teleported into the computation. To aid this protocol, we develop a new gate-efficient streaming version of quantum purity amplification that matches the optimal sample complexity in a wide range of parameters and is therefore of independent interest. The exponential reduction in fault-tolerant quantum resources comes at the expense of an exponential quantity of purely classical complexity-each of the n iterations of the protocol requires adaptively updating the $2^{n}$-size classical dataset and providing the noisy QRAM device with access to the updated dataset at the next iteration. We show that this classical operation can be parallelized to poly $(n)$ classical circuit depth, but only in a model where classical sparse matrix-vector multiplication for $2^{n}$-dimensional vectors can be as well. While our protocol demonstrates that QRAM is more compatible with fault-tolerant quantum computation than previously thought, the need for significant classical computational complexity exposes potentially fundamental limitations to realizing a truly poly $(n)$-cost faulttolerant QRAM.

A unitary k-design is an ensemble of unitaries that matches the first$k$moments of the Haar measure. In this work, we provide two efficient constructions of k-designs on n-qubits using new random matrix theory techniques. Our first construction is based on exponentiating sums of random i.i.d. Hermitian matrices and uses O(k2n2)-many gates. In the spirit of central limit theorems, we show that this random sum approximates the Gaussian Unitary Ensemble (GUE). We then show that the product of just two exponentiated GUE matrices is already approximately Haar random. Our second construction is based on products of exponentiated sums of random permutations and uses Õ($k$poly ($n$)) many gates. The$k$dependence is optimal (up to polylogarithmic factors) and is inherited from the efficiency of existing k-wise independent permutations. Furthermore, replacing random permutations with quantum-secure pseudorandom permutations (PRPs), we also obtain a pseudorandom unitary (PRU) ensemble that is secure under nonadaptive queries. A central feature of both proofs is a new connection between the polynomial method in quantum query complexity and the large-dimension ($N$) expansion in random matrix theory. In particular, the first construction uses the polynomial method to control high moments of certain random matrix ensembles without requiring delicate Weingarten calculations. In doing so, we define and solve a moment problem on the unit circle, asking whether a finite number of equally weighted points can reproduce a given set of moments. In our second construction, the key step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-$N$expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension$N$.

We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), Ising spin glasses (p-spin model), and k-local constraint satisfaction problems (k-CSP). We show that either (a) the algorithm finds the optimal solution in time O*(2(0.5−c)n) for an n-independent constant c, a 2cn advantage over Grover’s algorithm; or (b) there are sufficiently many low-cost solutions such that classical random guessing produces a (1−η) approximation to the optimal cost value in sub-exponential time for arbitrarily small choice of η. Additionally, we show that for a large fraction of random instances from the k-spin model and for any fully satisfiable or slightly frustrated k-CSP formula, statement (a) is the case. The algorithm and its analysis are largely inspired by Hastings’ short-path algorithm.

Recall the classical hypothesis testing setting with two sets of probability distributions P and Q. One receives either n i.i.d. samples from a distribution p E P or from a distribution q E Q and wants to decide from which set the points were sampled. It is known that the optimal exponential rate at which errors decrease can be achieved by a simple maximum-likelihood ratio test which does not depend on p or q, but only on the sets P and Q. We consider an adaptive generalization of this model where the choice of p E P and q E Q can change in each sample in some way that depends arbitrarily on the previous samples. In other words, in the kth round, an adversary, having observed all the previous samples in rounds 1, . . . , k - 1, chooses pk E P and qk E Q, with the goal of confusing the hypothesis test. We prove that even in this case, the optimal exponential error rate can be achieved by a simple maximum-likelihood test that depends only on P and Q. We then show that the adversarial model has applications in hypothesis testing for quantum states using restricted measurements. For example, it can be used to study the problem of distinguishing entangled states from the set of all separable states using only measurements that can be implemented with local operations and classical communication (LOCC). The basic idea is that in our setup, the deleterious effects of entanglement can be simulated by an adaptive classical adversary. We prove a quantum Stein's Lemma in this setting: In many circumstances, the optimal hypothesis testing rate is equal to an appropriate notion of quantum relative entropy between two states. In particular, our arguments yield an alternate proof of Li and Winter's recent strengthening of strong subadditivity for von Neumann entropy.

We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.

We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time n1/2m1/2s2poly(log(n), log(m), R, r, 1/δ), with n and s the dimension and row-sparsity of the input matrices, respectively, m the number of constraints, δ the accuracy of the solution, and R, r upper bounds on the size of the optimal primal and dual solutions, respectively. This gives a square-root unconditional speed-up over any classical method for solving SDPs both in n and m. We prove the algorithm cannot be substantially improved (in terms of n and m) giving a Ω(n1/2+ m2) quantum lower bound for solving semidefinite programs with constant s, R, r and δ. The quantum algorithm is constructed by a combination of quantum Gibbs sampling and the multiplicative weight method. In particular it is based on a classical algorithm of Arora and Kale for approximately solving SDPs. We present a modification of their algorithm to eliminate the need for solving an inner linear program which may be of independent interest.

The problem of device-independent randomness amplification against no-signaling adversaries has so far been studied under the assumption that the weak source of randomness is uncorrelated with the (quantum) devices used in the amplification procedure. In this paper, we relax this assumption, and reconsider the original protocol of Colbeck and Renner using a Santha-Vazirani (SV) source. To do so, we introduce an SV-like condition for devices, namely that any string of SV source bits remains weakly random conditioned upon any other bit string from the same SV source and the outputs obtained when this further string is input into the devices. Assuming this condition, we show that a quantum device using a singlet state to violate the chained Bell inequalities leads to full randomness in the asymptotic scenario of a large number of settings, for a restricted set of SV sources (with$0 \leq \varepsilon < (2^{(1/12)} - 1)/(2(2^{(1/12)} + 1)) \approx 0.0144$). We also study a device-independent protocol that allows for correlations between the sequence of boxes used in the protocol and the SV source bits used to choose the particular box from whose output the randomness is obtained. Assuming the SV-like condition for devices, we show that the honest parties can achieve amplification of the weak source, for the parameter range$0 \leq \varepsilon <0.0132$, against a class of attacks given as a mixture of product box sequences, made of extremal no-signaling boxes, with additional symmetry conditions. Composable security proof against this class of attacks is provided.

Recall the classical hypothesis testing setting with two convex sets of probability distributions P and Q. One receives either n i.i.d. samples from a distribution p ∈ P or from a distribution q ∈ Q and wants to decide from which set the points were sampled. It is known that the optimal exponential rate at which errors decrease can be achieved by a simple maximum-likelihood ratio test which does not depend on p or q, but only on the sets P and Q.

Quantum de Finetti theorems are a useful tool in the study of correlations in quantum multipartite states. In this paper we prove two new quantum de Finetti theorems, both showing that under tests formed by local measurements in each of the subsystems one can get a much improved error dependence on the dimension of the subsystems. We also obtain similar results for non-signaling probability distributions. We give the following applications of the results to quantum complexity theory, polynomial optimization, and quantum information theory: We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the assumption there is no subexponential-time algorithm for SAT. In the protocol a prover sends to a verifier √n polylog(n) unentangled quantum states, each composed of O(log(n)) qubits, as a proof of the satisfiability of a 3-SAT instance with n variables and O(n) clauses. The quantum verifier checks the validity of the proof by performing local measurements on each of the proofs and classically processing the outcomes. We show that any similar protocol with O(n1/2 - ε) qubits would imply a exp (n1 - 2ε polylog(n))-time algorithm for 3-SAT. We show that the maximum winning probability of free games (in which the questions to each prover are chosen independently) can be estimated by linear programming in time exp(O(log|Q| + log2|A|/ε2) ), with |Q| and |A| the question and answer alphabet sizes, respectively, matching the performance of a previously known algorithm due to Aaronson, Impagliazzo, Moshkovitz, and Shor. This result follows from a new monogamy relation for non-locality, showing that k-extendible non-signaling distributions give at most a O(k-1/2) advantage over classical strategies for free games. We also show that 3-SAT with n variables can be reduced to obtaining a constant error approximation of the maximum winning probability under entangled strategies of O(√n)-player one-round non-local games, in which only two players are selected to send O(√n)-bit messages. We show that the optimization of certain polynomials over the complex hypersphere can be performed in quasipolynomial time in the number of variables $n$ by considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy of semidefinite programs. This can be considered an analogue to the hypersphere of a similar known results for the simplex. As an application to entanglement theory, we find a quasipolynomial-time algorithm for deciding multipartite separability. We consider a quantum tomography result due to Aaronson -- showing that given an unknown n-qubit state one can perform tomography that works well for most observables by measuring only O(n) independent and identically distributed (i.i.d.) copies of the state -- and relax the assumption of having i.i.d copies of the state to merely the ability to select subsystems at random from a quantum multipartite state. The proofs of the new quantum de Finetti theorems are based on information theory, in particular on the chain rule of mutual information. The results constitute improvements and generalizations of a recent de Finetti theorem due to Brandao, Christandl and Yard.

The local Hamiltonian problem consists of estimating the ground-state energy (given by the minimum eigenvalue) of a local quantum Hamiltonian. It can be considered as a quantum generalization of constraint satisfaction problems (CSPs) and has a key role in quantum complexity theory, being the first and most natural QMA-complete problem known. An interesting regime for the local Hamiltonian problem is that of extensive error, where one is interested in estimating the mean ground-state energy to constant accuracy. The problem is NP-hard by the PCP theorem, but whether it is QMA-hard is an important open question in quantum complexity theory. A positive solution would represent a quantum analogue of the PCP theorem. A key feature that distinguishes quantum Hamiltonians from classical CSPs is that the solutions may involve complicated entangled states. In this paper, we demonstrate several large classes of Hamiltonians for which product (i.e. unentangled) states can approximate the ground state energy to within a small extensive error.

The entanglement cost of a quantum channel is the minimal rate at which entanglement (between sender and receiver) is needed in order to simulate many copies of a quantum channel in the presence of free classical communication. In this paper, we show how to express this quantity as a regularized optimization of the entanglement formation over states that can be generated between sender and receiver. Our formula is the channel analog of a well-known formula for the entanglement cost of quantum states in terms of the entanglement of formation and shares a similar relation to the recently shattered hope for additivity. The entanglement cost of a quantum channel can be seen as the analog of the quantum reverse Shannon theorem in the case where free classical communication is allowed. The techniques used in the proof of our result are then also inspired by a recent proof of the quantum reverse Shannon theorem and feature the one-shot formalism for quantum information theory, the postselection technique for quantum channels as well as Sion's minimax theorem. We discuss two applications of our result. First, we are able to link the security in the noisy-storage model to a problem of sending quantum rather than classical information through the adversary's storage device. This not only improves the range of parameters where security can be shown, but also allows us to prove security for storage devices for which no results were known before. Second, our result has consequences for the study of the strong converse quantum capacity. Here, we show that any coding scheme that sends quantum information through a quantum channel at a rate larger than the entanglement cost of the channel has an exponentially small fidelity.

Is there a meaningful quantum counterpart to public communication? We argue that it is the symmetric-side channel. This connection is partially motivated by recent work, where it was found that if a sender would like to communicate a secret message to a receiver through an insecure quantum channel using a shared quantum state as a key, then the insecure quantum channel is only ever used to simulate a symmetric-side channel. Here, we further show, in complete analogy to the role of public classical communication, that assistance by a symmetric-side channel makes equal the distillable entanglement, the recently introduced mutual independence, and a generalization of the latter, which quantifies the extent to which one of the parties can perform quantum privacy amplification. Symmetric-side channels, and the closely related erasure channel, have been recently harnessed to provide examples of superactivation of the quantum channel capacity. Our findings give new insight into this nonadditivity and its relation to quantum privacy. In particular, we show that single-copy superactivation protocols with the erasure channel, which encompasses all examples of nonadditivity of the quantum capacity found to date, can be understood as a conversion of mutual independence into distillable entanglement.

We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Then using a relative entropy version of the quantum asymptotic equipartition property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger ( ε-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot ε-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the ε-independent version of the relative entropy-QAEP, we can recover both the Holevo- Schumacher- Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.

A natural question in characterizing the information theoretic power of quantum channels is to ask at what rate entanglement is needed in order to asymptotically simulate a quantum channel in the presence of free classical communication. We call this the entanglement cost of a channel, and prove a formula describing it for all channels. We discuss two applications. Firstly, we are able to link the security in the noisy-storage model to a problem of sending quantum rather than classical information through the adversary's storage device. This not only greatly improves the range of parameters where security could be shown previously, but allows us to prove security for storage devices for which no non-trivial statements were known before. Secondly, our result has consequences for the study of the strong converse quantum capacity. Here, we show that any coding scheme that sends quantum information through a quantum channel at a rate larger than the entanglement cost of the channel has an exponentially small fidelity.

We study the computational complexity of approximating the 2-to-q norm of linear operators (defined as |A|2->q = maxv≠ 0|Av|q/|v|2) for q > 2, as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following: For any constant even integer q ≥ 4, a graph G is a small-set expander if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2->q norm. As a corollary, a good approximation to the 2->q norm will refute the Small-Set Expansion Conjecture --- a close variant of the UGC. We also show that such a good approximation can be obtained in exp(n2/q) time, thus obtaining a different proof of the known subexponential algorithm for Small-Set-Expansion. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy certify an upper bound on the 2->4 norm of the projector to low degree polynomials over the Boolean cube, as well certify the unsatisfiability of the "noisy cube" and "short code" based instances of Unique-Games considered by prior works. This improves on the previous upper bound of exp(logO(1) n) rounds (for the "short code"), as well as separates the "Sum of Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to require ω(1) rounds. We show reductions between computing the 2->4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory. Three corollaries are: (i) the 2->4 norm is NP-hard to approximate to precision inverse-polynomial in the dimension, (ii) the 2->4 norm does not have a good approximation (in the sense above) unless 3-SAT can be solved in time exp(√n poly log(n)), and (iii) known algorithms for the quantum separability problem imply a non-trivial additive approximation for the 2->4 norm.

We present a quasipolynomial-time algorithm for solving the weak membership problem for the convex set of separable, i.e. non-entangled, bipartite density matrices. The algorithm decides whether a density matrix is separable or whether it is ε-away from the set of the separable states in time exp(O(ε-2 log|A| log|B|)), where |A| and |B| are the local dimensions, and the distance is measured with either the Euclidean norm, or with the so-called LOCC norm. The latter is an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states, each shared by two parties, using any protocol formed by quantum local operations and classical communication (LOCC) between the parties. We also obtain improved algorithms for optimizing over the set of separable states and for computing the ground-state energy of mean-field Hamiltonians. The techniques we develop are also applied to quantum Merlin-Arthur games, where we show that multiple provers are not more powerful than a single prover when the verifier is restricted to LOCC protocols, or when the verification procedure is formed by a measurement of small Euclidean norm. This answers a question posed by Aaronson et al. (Theory of Computing 5, 1, 2009) and provides two new characterizations of the complexity class QMA, a quantum analog of NP.

We obtain expressions for the optimal rates of one-shot entanglement manipulation under operations which generate a negligible amount of entanglement. As the optimal rates for entanglement distillation and dilution in this paradigm, we obtain the max- and min-relative entropies of entanglement, the two logarithmic robustnesses of entanglement, and smoothed versions thereof. This gives a new operational meaning to these entanglement measures. Moreover, by considering the limit of many identical copies of the shared entangled state, we partially recover the recently found reversibility of entanglement manipulation under the class of operations which asymptotically do not generate entanglement.