Solving systems of linear equations is a fundamental problem, but it can be computationally intensive for classical algorithms in high dimensions. Existing quantum algorithms can achieve exponential speedups for the quantum linear system problem (QLSP) in terms of the problem dimension, but the advantage is bottlenecked by condition number of the coefficient matrix. In this work, we propose a new quantum algorithm for QLSP inspired by the classical proximal point algorithm (PPA). Our proposed method can be viewed as a meta-algorithm that allows inverting a modified matrix via an existing QLSP solver, thereby directly approximating the solution vector instead of approximating the inverse of the coefficient matrix. By carefully choosing the step size eta, the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that hindered the applicability of previous approaches. Importantly, this is the first iterative framework for QLSP where a tunable parameter eta and initialization x_0 allows controlling the trade-off between the runtime and approximation error.

The local Hamiltonian (LH) problem is the canonical QMA-complete problem introduced by Kitaev. In this paper, we show its hardness in a very strong sense: we show that the 3-local Hamiltonian problem on n qubits cannot be solved classically in time O(2^{(1-ε)n}) for any ε > 0 under the Strong Exponential-Time Hypothesis (SETH), and cannot be solved quantumly in time O(2^{(1-ε)n/2}) for any ε > 0 under the Quantum Strong Exponential-Time Hypothesis (QSETH). These lower bounds give evidence that the currently known classical and quantum algorithms for LH cannot be significantly improved. Furthermore, we are able to demonstrate fine-grained complexity lower bounds for approximating the quantum partition function (QPF) with an arbitrary constant relative error. Approximating QPF with relative error is known to be equivalent to approximately counting the dimension of the solution subspace of QMA problems. We show the SETH and QSETH hardness to estimate QPF with constant relative error. We then provide a quantum algorithm that runs in O(√{2ⁿ}) time for an arbitrary 1/poly(n) relative error, matching our lower bounds and improving the state-of-the-art algorithm by Bravyi, Chowdhury, Gosset, and Wocjan (Nature Physics 2022) in the low-temperature regime. To prove our fine-grained lower bounds, we introduce the first size-preserving circuit-to-Hamiltonian construction that encodes the computation of a T-time quantum circuit acting on N qubits into a (d+1)-local Hamiltonian acting on N+O(T^{1/d}) qubits. This improves the standard construction based on the unary clock, which uses N+O(T) qubits.

In recent years, achieving verifiable quantum advantage on a NISQ device has emerged as an important open problem in quantum information. The sampling-based quantum advantages are not known to have efficient verification methods. This article investigates the verification of quantum advantage from a cryptographic perspective. We establish a strong connection between the verifiability of quantum advantage and cryptographic and complexity primitives, including efficiently samplable, statistically far but computationally indistinguishable pairs of (mixed) quantum states ( EFI ), pseudorandom states ( PRS ), and variants of minimum circuit size problems ( MCSP ). Specifically, we prove that a) a sampling-based quantum advantage is either verifiable or can be used to build EFI and even PRS and b) polynomial-time algorithms for a variant of MCSP would imply efficient verification of quantum advantages. Our work shows that the quest for verifiable quantum advantages may lead to applications of quantum cryptography, and the construction of quantum primitives can provide new insights into the verifiability of quantum advantages.

We establish connections between state tomography, pseudorandomness, quantum state synthesis, and circuit lower bounds. In particular, let $\mathfrak C $ be a family of non-uniform quantum circuits of polynomial size and suppose that there exists an algorithm that, given copies of $\ket \psi$, distinguishes whether $\ket \psi$ is produced by $\mathfrak C$ or is Haar random, promised one of these is the case. For arbitrary fixed constant $c$, we show that if the algorithm uses at most $O\!\left(2^{n^c}\right)$ time and $2^{n^{0.99}}$ samples then $\mathsf{stateBQE} \not\subset \mathsf{state}\mathfrak{C}$. Here $\mathsf{stateBQE} \coloneqq \mathsf{stateBQTIME}\left[2^{O(n)}\right]$ and $\mathsf{state}\mathfrak{C}$ are state synthesis complexity classes as introduced by Rosenthal and Yuen (2022), which capture problems with classical inputs but quantum output. Note that efficient tomography implies a similarly efficient distinguishing algorithm against Haar random states, even for nearly exponential-time algorithms. Because every state produced by a polynomial-size circuit can be learned with $2^{O(n)}$ samples and time, or $\omega(\mathrm{poly}(n))$ samples and $2^{\omega(\mathrm{poly}(n))}$ time, we show that even slightly non-trivial quantum state tomography algorithms would lead to new statements about quantum state synthesis. Finally, a slight modification of our proof shows that distinguishing algorithms for quantum states can imply circuit lower bounds for decision problems as well. We then take these results and port them over to the setting of unitary learning and unitary synthesis. All combined, this helps shed light on why time-efficient tomography algorithms for non-uniform quantum circuit classes has only had limited and partial progress. Our work extends the results of Arunachalam et. al. (2022), which revealed a connection between quantum learning of \emph{Boolean functions} and circuit lower bounds for \emph{classical} circuit classes, to the setting of state (resp. unitary) tomography and state (resp. unitary) synthesis. As a result, we establish a conditional pseudorandom state (resp. unitary) generator, a circuit size hierarchy theorems for non-uniform state (resp. unitary) synthesis, and connections between state (resp. unitary) synthesis class separations and decision class separations, which may be of independent interest.

Quantum state learning is a fundamental problem in physics and computer science. As near-term quantum devices are error-prone, it is important to design error-resistant algorithms. Apart from device errors, other unexpected factors could also affect the algorithm, such as careless human read-out error, or even a malicious hacker deliberately altering the measurement results. Thus, we want our algorithm to work even in the worst case when things go against our favor.We consider the practical setting of single-copy measurements and propose the $\gamma$-adversarial corruption model where an imaginary adversary can arbitrarily change $\gamma$-fraction of the measurement outcomes. This is stronger than the $\gamma$-bounded SPAM noise model, where the post-measurement state changes by at most $\gamma$ in trace distance. Under our stronger model of corruption, we design an algorithm using non-adaptive measurements that can learn an unknown rank- r state up to $\tilde{O}(\gamma \sqrt{r})$ in trace distance, provided that the number of copies is sufficiently large. We further prove an information-theoretic lower bound of $\Omega(\gamma \sqrt{r})$ for non-adaptive measurements, demonstrating the optimality of our algorithm. Our upper and lower bounds also hold for quantum state testing, where the goal is to test whether an unknown state is equal to a given state or far from it. Our results are intriguingly optimistic and pessimistic at the same time. For general states, the error is dimension-dependent and $\gamma \sqrt{d}$ in the worst case, meaning that only corrupting a very small fraction $(1 / \sqrt{d})$ of the outcomes could totally destroy any non-adaptive learning algorithm. However, for constant-rank states that are useful in many quantum algorithms, it is possible to achieve dimension-independent error, even in the worst-case adversarial setting.

Hamiltonian simulation is one of the most important problems in the field of quantum computing. There have been extended efforts on designing algorithms for faster simulation, and the evolution time T for the simulation greatly affect algorithm runtime as expected. While there are some specific types of Hamiltonians that can be fast-forwarded, i.e., simulated within time o(T), for some large classes of Hamiltonians (e.g., all local/sparse Hamiltonians), existing simulation algorithms require running time at least linear in the evolution time T. On the other hand, while there exist lower bounds of Ω(T) circuit size for some large classes of Hamiltonian, these lower bounds do not rule out the possibilities of Hamiltonian simulation with large but "low-depth" circuits by running things in parallel. As a result, physical systems with system size scaling with T can potentially do a fast-forwarding simulation. Therefore, it is intriguing whether we can achieve fast Hamiltonian simulation with the power of parallelism. In this work, we give a negative result for the above open problem in various settings. In the oracle model, we prove that there are time-independent sparse Hamiltonians that cannot be simulated via an oracle circuit of depth o(T). In the plain model, relying on the random oracle heuristic, we show that there exist time-independent local Hamiltonians and time-dependent geometrically local Hamiltonians on n qubits that cannot be simulated via an oracle circuit of depth o(T/n^c), where the Hamiltonians act on n qubits, and c is a constant. Lastly, we generalize the above results and show that any simulators that are geometrically local Hamiltonians cannot do the simulation much faster than parallel quantum algorithms.

Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “ Any quantum polynomial-time algorithm can be implemented with only O (log n ) quantum depth interspersed with polynomial-time classical computations. ” This can be formalized as asserting the equivalence of BQP and “ BQNC BPP .” However, Aaronson conjectured that “ there exists an oracle separation between BQP and BPP BQNC . ” BQNC BPP and BPP BQNC are two natural and seemingly incomparable ways of hybrid classical-quantum computation. In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d , there exists an oracle that separates quantum depth d and 2 d +1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.

Despite potential quantum supremacy, state-of-the-art quantum neural networks (QNNs) suffer from low inference accuracy. First, the current Noisy Intermediate-Scale Quantum (NISQ) devices with high error rates of 10− 3 to 10− 2 significantly degrade the accuracy of a QNN. Second, although recently proposed Re-Uploading Units (RUUs) introduce some non-linearity into the QNN circuits, the theory behind it is not fully understood. Furthermore, previous RUUs that repeatedly upload original data can only provide marginal accuracy improvements. Third, current QNN circuit ansatz uses fixed two-qubit gates to enforce maximum entanglement capability, making task-specific entanglement tuning impossible, resulting in poor overall performance. In this paper, we propose a Quantum Multilayer Perceptron (QMLP) architecture featured by error-tolerant input embedding, rich nonlinearity, and enhanced variational circuit ansatz with parameterized two-qubit entangling gates. Compared to prior arts, QMLP increases the inference accuracy on the 10-class MNIST dataset by 10% with 2 × fewer quantum gates and 3 × reduced parameters. Our source code is available and can be found in https://github.com/chuchengc/QMLP/.

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all prior results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ 2 -norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

We investigate the existence of constant-round post-quantum black-box zero-knowledge protocols for NP. As a main result, we show that there is no constant-round post-quantum black-box zero-knowledge argument for NP unless$\text{NP} \subseteq \text{BQP}$. As constant-round black-box zero-knowledge arguments for NP exist in the classical setting, our main result points out a fundamental difference between post-quantum and classical zero-knowledge protocols. Combining previous results, we conclude that unless$\text{NP} \subseteq \text{BQP}$, constant-round post-quantum zero-knowledge protocols for NP exist if and only if we use non-black-box techniques or relax certain security requirements such as relaxing standard zero-knowledge to$\epsilon$-zero-knowledge. Additionally, we also prove that three-round and public-coin constant-round post-quantum black-box$\epsilon$-zero-knowledge arguments for NP do not exist unless$\text{NP} \subseteq \text{BQP}$.

The closest pair problem is a fundamental problem of computational geometry: given a set of n points in a d-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in O(n log n) time in constant dimensions (i.e., when d = O(1)). This paper asks and answers the question of the problem’s quantum time complexity. Specifically, we give an Õ(n^(2/3)) algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient history-independent data structure that supports quantum interference. In polylog(n) dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover’s algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum fine-grained complexity and introduce the Quantum Strong Exponential Time Hypothesis (QSETH), which is based on the assumption that Grover’s algorithm is optimal for CNF-SAT when the clause width is large. We show that the naïve Grover approach to closest pair in higher dimensions is optimal up to an n^o(1) factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.

We present classical sublinear-time algorithms for solving low-rank linear systems of equations. Our algorithms are inspired by the HHL quantum algorithm for solving linear systems and the recent breakthrough by Tang of dequantizing the quantum algorithm for recommendation systems. Let $A \in \mathbb{C}^{m \times n}$ be a rank-$k$ matrix, and $b \in \mathbb{C}^m$ be a vector. We present two algorithms: a "sampling" algorithm that provides a sample from $A^{-1}b$ and a "query" algorithm that outputs an estimate of an entry of $A^{-1}b$, where $A^{-1}$ denotes the Moore-Penrose pseudo-inverse. Both of our algorithms have query and time complexity $O(\mathrm{poly}(k, κ, \|A\|_F, 1/ε)\,\mathrm{polylog}(m, n))$, where $κ$ is the condition number of $A$ and $ε$ is the precision parameter. Note that the algorithms we consider are sublinear time, so they cannot write and read the whole matrix or vectors. In this paper, we assume that $A$ and $b$ come with well-known low-overhead data structures such that entries of $A$ and $b$ can be sampled according to some natural probability distributions. Alternatively, when $A$ is positive semidefinite, our algorithms can be adapted so that the sampling assumption on $b$ is not required.

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with $m$ constraint matrices, each of dimension $n$ and rank $r$, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time $O(m\cdot\mathrm{poly}(\log n,r,1/\varepsilon))$ given access to a sampling-based low-overhead data structure for the constraint matrices, where $\varepsilon$ is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application. Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest: $\bullet$ Weighted sampling: assuming sampling access to each individual constraint matrix $A_{1},\ldots,A_τ$, we propose a procedure that gives a good approximation of $A=A_{1}+\cdots+A_τ$. $\bullet$ Symmetric approximation: we propose a sampling procedure that gives the \emph{spectral decomposition} of a low-rank Hermitian matrix $A$. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.

Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation.

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffices to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

Prior research on viral marketing mostly focuses on promoting one single product item. In this work, we explore the idea of bundling multiple items for viral marketing and formulate a new research problem, called Bundle Configuration for SpreAd Maximization (BCSAM). Efficiently obtaining an optimal product bundle under the setting of BCSAM is very challenging. Aiming to strike a balance between the quality of solution and the computational overhead, we systematically explore various heuristics to develop a suite of algorithms, including κ-Bundle Configuration and Aggregated Bundle Configuration. Moreover, we integrate all the proposed ideas into one efficient algorithm, called Aggregated Bundle Configuration (ABC). Finally, we conduct an extensive performance evaluation on our proposals. Experimental results show that ABC significantly outperforms its counterpart and two baseline approaches in terms of both computational overhead and bundle quality.