Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra

TL;DR

The paper presents a qubit-efficient, oracle-free framework for sampling properties of matrix functions by starting from a classical Pauli-basis description of the matrix and using Fourier-series representations. It develops a general randomized sampling theorem that uses only $\log(N)+1$ qubits and scales with the Pauli weight $\lambda$ and Fourier parameters, enabling end-to-end tasks such as solving linear systems, estimating ground-state and Gibbs-state properties, and computing Green's functions without coherent data access. Concrete results include a randomized quantum linear-system solver, ground-state and Gibbs-state property estimation, and a Green's-function application, all with favorable space costs and competitive gate-depth under structural Pauli weight conditions. The work positions these qubit-efficient methods as viable candidates for early fault-tolerant quantum computing and provides detailed comparisons with traditional block-encoding and LCU-based approaches, highlighting how structural data representations can unlock practical quantum advantages. It also discusses classical sampling power under low Pauli weight and outlines several open questions for extending the Pauli-access paradigm to broader matrix-function problems.

Abstract

We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures. Our algorithms start from a classical data structure in which the matrix of interest is specified in the Pauli basis. For $N\times N$ Hermitian matrices, the space cost is $\log(N)+1$ qubits and depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size $O(N^2)$, when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as algorithms for sampling properties of ground states and Gibbs states of Hamiltonians. As a concrete application, we combine these sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.

Figures (3)

• Figure 1: Motivation of our work. We reduce quantum hardware requirements for quantum algorithms on classical data by removing the need for quantum data structures or quantum oracles. This is achieved by replacing coherent access to the data with a classical description of the data in the Pauli basis, and utilizing a randomized algorithm that samples the outputs of many quantum circuits. These circuits are chosen independently and thus in theory can also be parallelized, trading reduced total runtime for additional space cost in the form of many quantum processors. Our approach uses circuits with at most $\log N + 1$ qubits when processing data from $N \times N$ Hermitian matrices. This can be compared to other algorithms that utilize quantum data access models which may have significantly greater qubit overhead overall.
• Figure 2: Circuits used in our algorithms. We sample strings of quantum gates, consisting of specified Pauli operators and Pauli rotations, and perform quantum circuit runs with controlled versions of these gates. (a) Hadamard test circuit. Measuring the expectation value of $Z$ on the first register returns Re($\langle \psi |U| \psi \rangle$) and Im($\langle \psi |U| \psi \rangle$ for choices of $G = \mathbbm{1}$ and $G = S^{\dag}:=| \space 0 \rangle \langle 0 \space |-i| \space 1 \rangle \langle 1 \space |$ respectively. (b) Generalized Hadamard test circuit. Applying controlled-$U$ and anticontrolled-$V$, followed by measurement of the observable $X \otimes O$ yields $\frac{1}{2}\left(\langle \psi |U^{\dag}OV| \psi \rangle+ \langle \psi |V^{\dag}OU| \psi \rangle \right)$.
• Figure 3: Generalized Hadamard test. We use this circuit along with a modified measurement in Algorithm \ref{['alg:observable']} to sample from the solution vector. Rather than measuring the observable $Z \otimes O$ we measure all qubits in the computational basis.

Theorems & Definitions (53)

• Proposition 1: Sampling from Fourier series
• Theorem 1: Generalized sampling from Fourier approximations
• Remark : Sampling from output vector
• Proposition 2: Sampling normalization constant
• Proposition 3: Classical polynomial sampling
• Corollary 1: Linear systems
• Remark : Non-Hermitian matrices
• Proposition 4: Statistical encoding of input vector
• Corollary 2: Ground state property estimation
• Corollary 3: Gibbs state property estimation