[Paper Review] Estimating diagonal entries of powers of sparse symmetric matrices is BQP-complete

Dominik Janzing, Paweł Wocjan|ArXiv.org|Jun 27, 2006

Quantum Computing Algorithms and Architecture19 references3 citations

TL;DR

This paper proves that estimating diagonal entries of powers of sparse symmetric matrices is BQP-complete, establishing a direct link between quantum computational complexity and matrix analysis. The problem is efficiently solvable on a quantum computer using phase estimation and measurements, and it captures the full power of BQP by encoding quantum circuits into sparse matrices whose spectral moments distinguish between acceptance and rejection outcomes.

ABSTRACT

Let A be a real symmetric matrix of size N such that the number of the non-zero entries in each row is polylogarithmic in N and the positions and the values of these entries are specified by an efficiently computable function. We consider the problem of estimating an arbitrary diagonal entry (A^m)_jj of the matrix A^m up to an error of εb^m, where b is an a priori given upper bound on the norm of A, m and εare polylogarithmic and inverse polylogarithmic in N, respectively. We show that this problem is BQP-complete. It can be solved efficiently on a quantum computer by repeatedly applying measurements of A to the jth basis vector and raising the outcome to the mth power. Conversely, every quantum circuit that solves a problem in BQP can be encoded into a sparse matrix such that some basis vector |j> corresponding to the input induces two different spectral measures depending on whether the input is accepted or not. These measures can be distinguished by estimating the mth statistical moment for some appropriately chosen m, i.e., by the jth diagonal entry of A^m. The problem is still in BQP when generalized to off-diagonal entries and it remains BQP-hard if A has only -1, 0, and 1 as entries.

Motivation & Objective

• To characterize the complexity class BQP through a natural matrix problem involving sparse symmetric matrices.
• To show that estimating diagonal entries of matrix powers is both in BQP and BQP-hard under specific accuracy and sparsity constraints.
• To generalize the result to matrices with only entries in {−1, 0, 1}, demonstrating robustness of the BQP-completeness result.
• To establish a connection between quantum computation and spectral measures of sparse matrices, offering a new perspective on BQP's computational power.

Proposed method

• Use the quantum phase estimation algorithm to measure the spectral distribution of a sparse symmetric matrix A acting on a basis state |j⟩.
• Estimate the m-th statistical moment of the spectral measure by repeatedly measuring A on |j⟩ and averaging the m-th powers of outcomes.
• Encode a BQP quantum circuit into a sparse symmetric matrix A such that the spectral measure of |j⟩ depends on whether the input is accepted or rejected.
• Construct A as the real part of a unitary matrix to ensure operator norm ≤ 1, enabling bounded error scaling.
• Use a controlled construction of A with entries in {−1, 0, 1} by rescaling circuits composed of Toffoli and Hadamard gates, preserving BQP-hardness.
• Leverage the fact that the m-th moment (A^m)_{jj} differs significantly between accepted and rejected inputs when m is chosen as M^3 and M is large.

Experimental results

Research questions

• RQ1 Can estimating diagonal entries of powers of sparse symmetric matrices capture the full power of quantum computation?
• RQ2 Is this matrix estimation problem efficiently solvable on a quantum computer under polylogarithmic sparsity and accuracy constraints?
• RQ3 Does the BQP-completeness result persist when matrix entries are restricted to {−1, 0, 1}?
• RQ4 Can the spectral moments of a sparse matrix's action on a basis state be used to simulate any BQP decision problem?
• RQ5 What is the relationship between the structure of sparse matrices and the complexity of their spectral moment estimation?

Key findings

• The problem of estimating (A^m)_{jj} up to error εb^m, where b is the operator norm of A, is BQP-complete under the given constraints on sparsity and accuracy.
• The estimation can be performed efficiently on a quantum computer using repeated measurements and phase estimation to extract the m-th moment of the spectral measure.
• The problem remains BQP-hard even when A has only entries in {−1, 0, 1}, showing the result is robust to entry-value restrictions.
• For a sufficiently large m = M^3, the diagonal entry (A^m)_{jj} exceeds 1/(4M) when the input is not in the language L, and is less than −1/(4M) when it is in L, enabling decision via thresholding.
• The construction ensures that the spectral measure of |j⟩ depends on the quantum circuit’s acceptance outcome, allowing the diagonal entry to encode the computational result.