Quantum computational complexity of matrix functions (2410.13937v2)

Abstract: We investigate the dividing line between classical and quantum computational power in estimating properties of matrix functions. More precisely, we study the computational complexity of two primitive problems: given a function $f$ and a Hermitian matrix $A$, compute a matrix element of $f(A)$ or compute a local measurement on $f(A)|0\rangle^{\otimes n}$, with $|0\rangle^{\otimes n}$ an $n$-qubit reference state vector, in both cases up to additive approximation error. We consider four functions -- monomials, Chebyshev polynomials, the time evolution function, and the inverse function -- and probe the complexity across a broad landscape covering different problem input regimes. Namely, we consider two types of matrix inputs (sparse and Pauli access), matrix properties (norm, sparsity), the approximation error, and function-specific parameters. We identify BQP-complete forms of both problems for each function and then toggle the problem parameters to easier regimes to see where hardness remains, or where the problem becomes classically easy. As part of our results, we make concrete a hierarchy of hardness across the functions; in parameter regimes where we have classically efficient algorithms for monomials, all three other functions remain robustly BQP-hard, or hard under usual computational complexity assumptions. In identifying classically easy regimes, among others, we show that for any polynomial of degree $\mathrm{poly}(n)$ both problems can be efficiently classically simulated when $A$ has $O(\log n)$ non-zero coefficients in the Pauli basis. This contrasts with the fact that the problems are BQP-complete in the sparse access model even for constant row sparsity, whereas the stated Pauli access efficiently constructs sparse access with row sparsity $O(\log n)$. Our work provides a catalog of efficient quantum and classical algorithms for fundamental linear-algebra tasks.

• The paper establishes a complexity hierarchy by demonstrating that while some matrix functions like monomials are computable classically, functions such as Chebyshev polynomials and matrix inversion remain BQP-hard.
• The paper employs detailed reductions and explores varied input models, including sparse and Pauli access, to rigorously analyze computational challenges.
• The paper identifies scenarios of classical simulability that guide algorithm design and highlight key conditions for achieving quantum computational advantage.

Quantum Computational Complexity of Matrix Functions

The investigation into the quantum computational complexity of matrix functions has captured significant attention, specifically regarding the delineation between classical and quantum computational power. In the depicted paper, the authors explore the complexity of estimating properties of matrix functions, focusing on two primitive problems: computing a matrix element of a function of a Hermitian matrix, and performing a local measurement on a matrix function. These are examined from a quantum computational perspective.

Key Findings and Methodology

The research addresses four specific functions: monomials, Chebyshev polynomials, the time evolution function, and the inverse function. The investigation covers varying input regimes of Hermitian matrices, including sparse and Pauli access models, different matrix properties like norm and sparsity, approximation errors, and function-specific parameters.

1. Complexity Hierarchy:
   • The authors establish that for monomials, there are efficient classical algorithms in certain regimes, such as when matrices have a certain sparsity or are normalized under specific norms.
   • Conversely, other functions like Chebyshev polynomials, time evolution, and matrix inversion maintain BQP-hardness even under restricted input conditions, marking a complexity hierarchy.
2. BQP-completeness:
   • The paper identifies BQP-complete forms for the problems associated with each function. Particularly, for monomials and Chebyshev polynomials, complexity remains robustly intractable for quantum computations even with fixed precision.
   • Utilizing reductions from problems like BQPCircuitSimulation, the paper solidifies BQP-completeness through notable constructions such as the Hamiltonian time evolution and matrix inversion.
3. Classical Simulability:
   • Through a series of lemmas and propositions, the authors outline scenarios where classical algorithms can approximate or compute certain matrix functions efficiently. For instance, when matrices fulfill conditions such as sparsity or have a limited number of non-zero coefficients in the Pauli basis.
4. Effect of Access Models:
   • A novel insight from the paper is the difference in computational complexity emerging from sparse versus Pauli access models. While computational hardness is prevalent in sparse access, some problems become tractable under Pauli access, highlighting the importance of matrix representation in complexity considerations.

Implications and Future Directions

The implications of this work are broad, affecting both theoretical and practical domains of quantum computing:

• Theoretical Insights: The work contributes to an understanding of where quantum advantage is fundamentally possible by explicitly characterizing the complexity of matrix function problems across different scenarios. This helps demarcate the boundaries of quantum versus classical capabilities.
• Algorithmic Development: Insights into classical simulability conditions inform the development of algorithms that are efficient on classical machines while also accentuating areas where quantum computers offer genuine advantages.
• Access Model Exploration: The difference between matrix access models advises ongoing research into how quantum algorithms can utilize specific structures in data representation to achieve computational benefits.

In terms of future developments, further research could extend to different classes of functions or more nuanced access models. Understanding how these variations impact computational complexity could fine-tune the identification of problems suitable for quantum computation.

Conclusion

The paper offers a comprehensive paper on matrix function complexity, providing both a theoretical framework for understanding the difficulty of these computations and practical guidelines for algorithm design. The work highlights specific scenarios leveraging quantum advantages, thus paving the way for future exploration into bridging classical and quantum computation gaps.