arXiv subjects

Changpeng Shao

Publications and source records attributed to Changpeng Shao.

At least 19 recordsLinked to original sources

arXiv2026

Low-ancilla block encodings via Hamiltonian simulation

Block encodings are a central primitive in quantum algorithms, but standard constructions typically require logarithmic ancilla overhead and complicated controlled operations. Recent lower bounds further show that such ancilla overhead is unavoidable for exact constructions in broad circuit models. We show that this barrier can be bypassed in the approximate setting. Specifically, we present a simple single-ancilla construction that converts Hamiltonian evolution into a block encoding of the underlying Hamiltonian, via generalized quantum signal processing. For operators given by Hermitian decompositions $A=\sum_{j=1}^L \alpha_j H_j$, we instantiate this block-encoding construction in two ways, which differ in how the required Hamiltonian evolution is implemented. Using higher-order Trotterization, we obtain an $\varepsilon$-approximate block encoding of $A$ with only one ancilla qubit and circuit depth $\widetilde O\big(L(\alpha/\varepsilon)^{o(1)}\big),$ where $\alpha=\sum_j \alpha_j$. Using multiproduct formulas, we obtain circuit depth $\widetilde O(L)$, at the cost of $O(\log\log(1/\varepsilon))$ ancilla qubits. Our constructions provide alternatives to the standard LCU framework, with a focus on reducing the number of ancilla qubits while maintaining (near-)optimal circuit depth.

quant-ph↗

arXiv2026

Exponential Lindbladian fast forwarding and exponential amplification of certain Gibbs state properties

Fast-forwarding refers to the ability to simulate a system of time $t$ using significantly fewer than $t$ queries or circuit depth. While various Hamiltonian systems are known to circumvent the no fast-forwarding theorem, analogous results for dissipative dynamics, governed by Lindbladians, remain largely unexplored. We first present a quantum algorithm for simulating purely dissipative Lindbladians with unitary jump operators, achieving additive query complexity $\mathcal{O}\left(t + \log(\varepsilon^{-1})\right)$ up to error~$\varepsilon$, improving previous algorithms. When the jump operators have certain structures (i.e., block-diagonal Paulis), the algorithm can be modified to achieve exponential fast-forwarding, attaining circuit depth $\mathcal{O}\left(\log\left(t + \log(\varepsilon^{-1})\right)\right)$, while preserving query complexity via parallel access. Using these fast-forwarding techniques, we develop a quantum algorithm for estimating Gibbs state properties of the form $\langle ψ_1 | e^{-β(H + I)} | ψ_2 \rangle$, up to additive error $ε$, with $H$ the Hamiltonian and $β$ the inverse temperature. For input states exhibiting certain coherence conditions -- e.g.,~$\langle 0|^{\otimes n} e^{-β(H + I)} |+\rangle^{\otimes n}$ -- our method achieves exponential improvement in complexity (measured by circuit depth), $\mathcal{O} (2^{-n/2} ε^{-1} \log β),$ compared to the quantum singular value transformation-based approach, with complexity $\tilde{\mathcal{O}} (ε^{-1} \sqrtβ )$. We show how to apply this exponential improvement to applications such as the ground state overlap testing and amplitude estimation. For general $| ψ_1 \rangle$ and $| ψ_2 \rangle$, we also show how the level of improvement is changed with the coherence resource in $| ψ_1 \rangle$ and $| ψ_2 \rangle$.

quant-ph↗

arXiv2026

Quantum spectral method for gradient and Hessian estimation

Gradient descent is one of the most basic algorithms for solving continuous optimization problems. In [Jordan, PRL, 95(5):050501, 2005], Jordan proposed the first quantum algorithm for estimating gradients of functions close to linear, with exponential speedup in the black-box model. This quantum algorithm was greatly enhanced and developed by [Gilyén, Arunachalam, and Wiebe, SODA, pp. 1425-1444, 2019], providing a quantum algorithm with optimal query complexity $\widetildeΘ(\sqrt{d}/\varepsilon)$ for a class of smooth functions of $d$ variables, where $\varepsilon$ is the accuracy. This is quadratically faster than classical algorithms for the same problem. In this work, we continue this research by proposing a new quantum algorithm for another class of functions, namely, analytic functions $f(\boldsymbol{x})$ which are well-defined over the complex field. Given phase oracles to query the real and imaginary parts of $f(\boldsymbol{x})$ respectively, we propose a quantum algorithm that returns an $\varepsilon$-approximation of its gradient with query complexity $\widetilde{O}(1/\varepsilon)$. As an extension, we also propose two quantum algorithms for Hessian estimation, aiming to improve quantum analogs of Newton's method. The two algorithms have query complexity $\widetilde{O}(d/\varepsilon)$ and $\widetilde{O}(d^{1.5}/\varepsilon)$, respectively, under different assumptions. Moreover, if the Hessian is promised to be $s$-sparse, we then have two new quantum algorithms with query complexity $\widetilde{O}(s/\varepsilon)$ and $\widetilde{O}(sd/\varepsilon)$, respectively. We also prove a lower bound of $\widetildeΩ(d)$ for Hessian estimation in the general case.

quant-ph↗

arXiv2026

Worst-case Harrow-Hassidim-Lloyd algorithm with average-case correct quantum Fourier transform

In [\href{https://quantum-journal.org/papers/q-2022-12-07-872/}{Quantum 6, 872, 2022}], Linden and de Wolf proposed a lightweight protocol for verifying average-case correctness of the quantum Fourier transform (QFT). They showed that good average-case QFT performance is sufficient for good worst-case performance in several quantum information-processing tasks. In this work, we study whether such average-case guarantees are also sufficient when the QFT is used coherently inside the Harrow--Hassidim--Lloyd algorithm. We show that the original average-case condition is not quite strong enough for this purpose, due to possible relative phase errors between different eigenspaces. To address this, we introduce a strengthened Linden--de Wolf-type verification condition that controls the relevant coherences, and prove that it guarantees worst-case correctness of the HHL algorithm in several natural settings.

quant-ph↗

arXiv2026

DQC1-completeness of normalized trace estimation for functions of log-local Hamiltonians

We study the computational complexity of estimating the normalized trace $2^{-n}Tr[f(A)]$ for a log-local Hamiltonian $A$ acting on $n$ qubits. This problem arises naturally in the DQC1 model, yet its complexity is only understood for a limited class of functions $f(x)$. We show that if $f(x)$ is a continuous function with approximate degree $\Omega({\rm poly}(n))$, then estimating $2^{-n}Tr[f(A)]$ up to constant additive error is DQC1-complete, under a technical condition on the polynomial approximation error of $f(x)$. This condition holds for a broad class of functions, including exponentials, trigonometric functions, logarithms, and inverse-type functions. We further prove that when $A$ is sparse, the classical query complexity of this problem is exponential in the approximate degree, assuming a conjectured lower bound for a trace variant of the $k$-Forrelation problem in the DQC1 query model. Together, these results identify the approximate degree as the key parameter governing the complexity of normalized trace estimation: it characterizes both the quantum complexity (via efficient DQC1 algorithms) and, conditionally, the classical hardness, yielding an exponential quantum-classical separation. Our proof develops a unified framework that cleanly combines circuit-to-Hamiltonian constructions, periodic Jacobi operators, and tools from polynomial approximation theory, including the Chebyshev equioscillation theorem.

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arXiv2025

Randomized Quantum Singular Value Transformation

We introduce the first randomized algorithms for Quantum Singular Value Transformation (QSVT), a unifying framework for many quantum algorithms. Standard implementations of QSVT rely on block encodings of the Hamiltonian, which are costly to construct, requiring a logarithmic number of ancilla qubits, intricate multi-qubit control, and circuit depth scaling linearly with the number of Hamiltonian terms. In contrast, our algorithms use only a single ancilla qubit and entirely avoid block encodings. We develop two methods: (i) a direct randomization of QSVT, where block encodings are replaced by importance sampling, and (ii) an approach that integrates qDRIFT into the generalized quantum signal processing framework, with the dependence on precision exponentially improved through classical extrapolation. Both algorithms achieve gate complexity independent of the number of Hamiltonian terms, a hallmark of randomized methods, while incurring only quadratic dependence on the degree of the target polynomial. We identify natural parameter regimes where our methods outperform even standard QSVT, making them promising for early fault-tolerant quantum devices. We also establish a fundamental lower bound showing that the quadratic dependence on the polynomial degree is optimal within this framework. We apply our framework to two fundamental tasks: solving quantum linear systems and estimating ground-state properties of Hamiltonians, obtaining polynomial advantages over prior randomized algorithms. Finally, we benchmark our ground-state property estimation algorithm on electronic structure Hamiltonians and the transverse-field Ising model with long-range interactions. In both cases, our approach outperforms prior work by several orders of magnitude in circuit depth, establishing randomized QSVT as a practical and resource-efficient alternative for early fault-tolerant quantum devices.

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arXiv2025

Quantum singular value transformation without block encodings: Near-optimal complexity with minimal ancilla

We develop new algorithms for Quantum Singular Value Transformation (QSVT), a unifying framework that encapsulates most known quantum algorithms and serves as the foundation for new ones. Existing implementations of QSVT rely on block encoding, incurring an intrinsic $O(\log L)$ ancilla overhead and circuit depth $\widetilde{O}(L dλ)$ for polynomial transformations of a Hamiltonian $H=\sum_{k=1}^L H_k$, where $d$ is the polynomial degree and $λ=\sum_{k}\|H_k\|$. We introduce a simple yet powerful approach that utilizes only basic Hamiltonian simulation techniques, namely, Trotter methods, to: (i) eliminate the need for block encoding, (ii) reduce the ancilla overhead to only a single qubit, and (iii) still maintain near-optimal complexity. Our method achieves a circuit depth of $\widetilde{O}(L(dλ_{\mathrm{comm}})^{1+o(1)})$, without requiring any complicated multi-qubit controlled gates. Moreover, $λ_{\mathrm{comm}}$ depends on the nested commutators of the terms of $H$ and can be substantially smaller than $λ$ for many physically relevant Hamiltonians, a feature absent in standard QSVT. To achieve these results, we make use of Richardson extrapolation in a novel way, systematically eliminating errors in any interleaved sequence of arbitrary unitaries and Hamiltonian evolution operators, thereby establishing a general framework that encompasses QSVT but is more broadly applicable. As applications, we develop end-to-end quantum algorithms for solving linear systems and estimating ground state properties of Hamiltonians, both achieving near-optimal complexity without relying on oracular access. Overall, our results establish a new framework for quantum algorithms, significantly reducing hardware overhead while maintaining near-optimal performance, with implications for both near-term and fault-tolerant quantum computing.

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arXiv2025

Testing quantum satisfiability

Quantum k-SAT (the problem of determining whether a k-local Hamiltonian is frustration-free) is known to be QMA_1-complete for k >= 3, and hence likely hard for quantum computers to solve. Building on a classical result of Alon and Shapira, we show that quantum k-SAT can be solved in randomised polynomial time given the `property testing' promise that the instance is either satisfiable (by any state) or far from satisfiable by a product state; by `far from satisfiable by a product state' we mean that εn^k constraints must be removed before a product state solution exists, for some fixed ε> 0. The proof has two steps: we first show that for a satisfiable instance of quantum k-SAT, most subproblems on a constant number of qubits are satisfiable by a product state. We then show that for an instance of quantum k-SAT which is far from satisfiable by a product state, most subproblems are unsatisfiable by a product state. Given the promise, quantum k-SAT may therefore be solved by checking satisfiability by a product state on randomly chosen subsystems of constant size.

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arXiv2025

Normalization of Quaternionic Polynomials in Coordinate-Free Quaternionic Variables in Conjugate-Alternating Order

Quaternionic polynomials occur naturally in applications of quaternions in science and engineering, and normalization of quaternionic polynomials is a basic manipulation. Once a Groebner basis is certified for the defining ideal I of the quaternionic polynomial algebra, the normal form of a quaternionic polynomial can be computed by routine top reduction with respect to the Groebner basis. In the literature, a Groebner basis under the conjugate-alternating order of quaternionic variables was conjectured for I in 2013, but no readable and convincing proof was found. In this paper, we present the first readable certification of the conjectured Groebner basis. The certification is based on several novel techniques for reduction in free associative algebras, which enables to not only make reduction to S-polynomials more efficiently, but also reduce the number of S-polynomials needed for the certification.

cs.SC↗

arXiv2025

Quantum and classical query complexities of functions of matrices

Let $A$ be an $s$-sparse Hermitian matrix, $f(x)$ be a univariate function, and $i, j$ be two indices. In this work, we investigate the query complexity of approximating $\bra{i} f(A) \ket{j}$. We show that for any continuous function $f(x):[-1,1]\rightarrow [-1,1]$, the quantum query complexity of computing $\bra{i} f(A) \ket{j}\pm \varepsilon/4$ is lower bounded by $Ω(\widetilde°_\varepsilon(f))$. The upper bound is at most quadratic in $\widetilde°_\varepsilon(f)$ and is linear in $\widetilde°_\varepsilon(f)$ under certain mild assumptions on $A$. Here the approximate degree $\widetilde°_\varepsilon(f)$ is the minimum degree such that there is a polynomial of that degree approximating $f$ up to additive error $\varepsilon$ in the interval $[-1,1]$. We also show that the classical query complexity is lower bounded by $\widetildeΩ((s/2)^{(\widetilde°_{2\varepsilon}(f)-1)/6})$ for any $s\geq 4$. Our results show that the quantum and classical separation is exponential for any continuous function of sparse Hermitian matrices, and also imply the optimality of implementing smooth functions of sparse Hermitian matrices by quantum singular value transformation. As another hardness result, we show that entry estimation problem (i.e., deciding $\bra{i} f(A) \ket{j}\geq \varepsilon$ or $\bra{i} f(A) \ket{j}\leq -\varepsilon$) is BQP-complete for any continuous function $f(x)$ as long as its approximate degree is large enough. The main techniques we used are the dual polynomial method for functions over the reals, linear semi-infinite programming, and tridiagonal matrices.

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arXiv2024

Lower bounds for quantum-inspired classical algorithms via communication complexity

Quantum-inspired classical algorithms provide us with a new way to understand the computational power of quantum computers for practically-relevant problems, especially in machine learning. In the past several years, numerous efficient algorithms for various tasks have been found, while an analysis of lower bounds is still missing. Using communication complexity, in this work we propose the first method to study lower bounds for these tasks. We mainly focus on lower bounds for solving linear regressions, supervised clustering, principal component analysis, recommendation systems, and Hamiltonian simulations. For those problems, we prove a quadratic lower bound in terms of the Frobenius norm of the underlying matrix. As quantum algorithms are linear in the Frobenius norm for those problems, our results mean that the quantum-classical separation is at least quadratic. As a generalisation, we extend our method to study lower bounds analysis of quantum query algorithms for matrix-related problems using quantum communication complexity. Some applications are given.

quant-ph↗

arXiv2024

Low-degree approximation of QAC$^0$ circuits

QAC$^0$ is the class of constant-depth quantum circuits with polynomially many ancillary qubits, where Toffoli gates on arbitrarily many qubits are allowed. In this work, we show that the parity function cannot be computed in QAC$^0$, resolving a long-standing open problem in quantum circuit complexity more than twenty years old. As a result, this proves ${\rm QAC}^0 \subsetneqq {\rm QAC}_{\rm wf}^0$. We also show that any QAC circuit of depth $d$ that approximately computes parity on $n$ bits requires $2^{\widetildeΩ(n^{1/d})}$ ancillary qubits, which is close to tight. This implies a similar lower bound on approximately preparing cat states using QAC circuits. Finally, we prove a quantum analog of the Linial-Mansour-Nisan theorem for QAC$^0$. This implies that, for any QAC$^0$ circuit $U$ with $a={\rm poly}(n)$ ancillary qubits, and for any $x\in\{0,1\}^n$, the correlation between $Q(x)$ and the parity function is bounded by ${1}/{2} + 2^{-\widetildeΩ(n^{1/d})}$, where $Q(x)$ denotes the output of measuring the output qubit of $U|x,0^a\rangle$. All the above consequences rely on the following technical result. If $U$ is a QAC$^0$ circuit with $a={\rm poly}(n)$ ancillary qubits, then there is a distribution $\mathcal{D}$ of bounded polynomials of degree polylog$(n)$ such that with high probability, a random polynomial from $\mathcal{D}$ approximates the function $\langle x,0^a| U^†Z_{n+1} U |x,0^a\rangle$ for a large fraction of $x\in \{0,1\}^n$. This result is analogous to the Razborov-Smolensky result on the approximation of AC$^0$ circuits by random low-degree polynomials.

quant-ph↗

arXiv2024

Reflective block Kaczmarz algorithms for least squares

In [Steinerberger, Q. Appl. Math., 79:3, 419-429, 2021] and [Shao, SIAM J. Matrix Anal. Appl. 44(1), 212-239, 2023], two new types of Kaczmarz algorithms, which share some similarities, for consistent linear systems were proposed. These two algorithms not only compete with many previous Kaczmarz algorithms but, more importantly, reveal some interesting new geometric properties of solutions to linear systems that are not obvious from the standard viewpoint of the Kaczmarz algorithm. In this paper, we comprehensively study these two algorithms. First, we theoretically analyse the algorithms given in [Steinerberger, Q. Appl. Math., 79:3, 419-429, 2021] for solving least squares. Second, we extend the two algorithms to block versions and provide their theoretical convergence rates. Our numerical experiments also verify the efficiency of these algorithms. Third, as a theoretical complement, we address some key questions left unanswered in [Shao, SIAM J. Matrix Anal. Appl. 44(1), 212-239, 2023].

math.NA↗

arXiv2024

Quantum algorithms for spectral sums

We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. The spectral sum of an PSD matrix $A$, for a function $f$, is defined as $ \text{Tr}[f(A)] = \sum_j f(λ_j)$, where $λ_j$ are the eigenvalues of $A$. Typical examples of spectral sums are the von Neumann entropy, the trace of $A^{-1}$, the log-determinant, and the Schatten $p$-norm, where the latter does not require the matrix to be PSD. The current best classical randomized algorithms estimating these quantities have a runtime that is at least linearly in the number of nonzero entries of the matrix and quadratic in the estimation error. Assuming access to a block-encoding of a matrix, our algorithms are sub-linear in the matrix size, and depend at most quadratically on other parameters, like the condition number and the approximation error, and thus can compete with most of the randomized and distributed classical algorithms proposed in the literature, and polynomially improve the runtime of other quantum algorithms proposed for the same problems. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory: approximating the number of triangles, the effective resistance, and the number of spanning trees within a graph.

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arXiv2023

Quantum speedup of leverage score sampling and its application

Leverage score sampling is crucial to the design of randomized algorithms for large-scale matrix problems, while the computation of leverage scores is a bottleneck of many applications. In this paper, we propose a quantum algorithm to accelerate this useful method. The speedup is at least quadratic and could be exponential for well-conditioned matrices. We also prove some quantum lower bounds, which suggest that our quantum algorithm is close to optimal. As an application, we propose a new quantum algorithm for rigid regression problems with vector solution outputs. It achieves polynomial speedups over the best classical algorithm known. In this process, we give an improved randomized algorithm for rigid regression.

quant-ph↗

arXiv2023

A deterministic Kaczmarz algorithm for solving linear systems

We propose a new deterministic Kaczmarz algorithm for solving consistent linear systems $A\mathbf{x}=\mathbf{b}$. Basically, the algorithm replaces orthogonal projections with reflections in the original scheme of Stefan Kaczmarz. Building on this, we give a geometric description of solutions of linear systems. Suppose $A$ is $m\times n$, we show that the algorithm generates a series of points distributed with patterns on an $(n-1)$-sphere centered on a solution. These points lie evenly on $2m$ lower-dimensional spheres $\{§_{k0},§_{k1}\}_{k=1}^m$, with the property that for any $k$, the midpoint of the centers of $§_{k0},§_{k1}$ is exactly a solution of $A\mathbf{x}=\mathbf{b}$. With this discovery, we prove that taking the average of $O(η(A)\log(1/\varepsilon))$ points on any $§_{k0}\cup§_{k1}$ effectively approximates a solution up to relative error $\varepsilon$, where $η(A)$ characterizes the eigengap of the orthogonal matrix produced by the product of $m$ reflections generated by the rows of $A$. We also analyze the connection between $η(A)$ and $κ(A)$, the condition number of $A$. In the worst case $η(A)=O(κ^2(A)\log m)$, while for random matrices $η(A)=O(κ(A))$ on average. Finally, we prove that the algorithm indeed solves the linear system $A^T W^{-1}A \mathbf{x} = A^T W^{-1} \mathbf{b}$, where $W$ is the lower-triangular matrix such that $W+W^T = 2AA^T$. The connection between this linear system and the original one is studied. The numerical tests indicate that this new Kaczmarz algorithm has comparable performance to randomized (block) Kaczmarz algorithms.

math.NA↗

arXiv2023

Quantum communication complexity of linear regression

Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases -- such as for low-rank matrices -- dequantized algorithms demonstrate that there cannot be an exponential quantum speedup. In this work, we show that quantum computers have provable polynomial and exponential speedups in terms of communication complexity for some fundamental linear algebra problems \update{if there is no restriction on the rank}. We mainly focus on solving linear regression and Hamiltonian simulation. In the quantum case, the task is to prepare the quantum state of the result. To allow for a fair comparison, in the classical case, the task is to sample from the result. We investigate these two problems in two-party and multiparty models, propose near-optimal quantum protocols and prove quantum/classical lower bounds. In this process, we propose an efficient quantum protocol for quantum singular value transformation, which is a powerful technique for designing quantum algorithms. This will be helpful in developing efficient quantum protocols for many other problems.

quant-ph↗

arXiv2023

Faster quantum-inspired algorithms for solving linear systems

We establish an improved classical algorithm for solving linear systems in a model analogous to the QRAM that is used by quantum linear solvers. Precisely, for the linear system $A\x = \b$, we show that there is a classical algorithm that outputs a data structure for $\x$ allowing sampling and querying to the entries, where $\x$ is such that $\|\x - A^{+}\b\|\leq ε\|A^{+}\b\|$. This output can be viewed as a classical analogue to the output of quantum linear solvers. The complexity of our algorithm is $\widetilde{O}(κ_F^4 κ^2/ε^2 )$, where $κ_F = \|A\|_F\|A^{+}\|$ and $κ= \|A\|\|A^{+}\|$. This improves the previous best algorithm [Gily{é}n, Song and Tang, arXiv:2009.07268] of complexity $\widetilde{O}(κ_F^6 κ^6/ε^4)$. Our algorithm is based on the randomized Kaczmarz method, which is a particular case of stochastic gradient descent. We also find that when $A$ is row sparse, this method already returns an approximate solution $\x$ in time $\widetilde{O}(κ_F^2)$, while the best quantum algorithm known returns $\ket{\x}$ in time $\widetilde{O}(κ_F)$ when $A$ is stored in the QRAM data structure. As a result, assuming access to QRAM and if $A$ is row sparse, the speedup based on current quantum algorithms is quadratic.

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