[Paper Review] Quantum Computation and Quadratically Signed Weight Enumerators

Emanuel Knill, Raymond Laflamme|ArXiv.org|Sep 30, 1999

Complexity and Algorithms in Graphs4 citations

TL;DR

This paper establishes a connection between quantum computation and the estimation of quadratically signed weight enumerators (QSWEs), proving that quantum computation is polynomially equivalent to classical probabilistic computation with an oracle for estimating these sums. It identifies a canonical BQP-complete problem and introduces new complexity classes linked to quantum computation through QSWEs, highlighting their role in distinguishing quantum from classical computational power.

ABSTRACT

We prove that quantum computation is polynomially equivalent to classical probabilistic computation with an oracle for estimating the value of simple sums, quadratically signed weight enumerators. The problem of estimating these sums can be cast in terms of promise problems and has two interesting variants. An oracle for the unconstrained variant may be more powerful than quantum computation, while an oracle for a more constrained variant is efficiently solvable in the one-bit model of quantum computation. Thus, problems involving estimation of quadratically signed weight enumerators yield problems in BQP (bounded error quantum polynomial time) that are distinct from the ones studied so far, include a canonical BQP complete problem, and can be used to define and study complexity classes and their relationships to quantum computation.

Motivation & Objective

• To establish a formal equivalence between quantum computation and classical probabilistic computation enhanced with an oracle for estimating quadratically signed weight enumerators (QSWEs).
• To identify and characterize new complexity classes related to quantum computation through the lens of QSWEs.
• To explore the computational power of oracles for unconstrained and constrained variants of QSWE estimation.
• To define a canonical BQP-complete problem based on QSWE estimation.
• To clarify the relationship between quantum computation and classical complexity classes via QSWE oracles.

Proposed method

• The authors define quadratically signed weight enumerators (QSWEs) as specific types of signed sums over binary vectors, tied to classical codes and quantum states.
• They introduce a promise problem variant of QSWE estimation, distinguishing between unconstrained and constrained forms.
• The paper proves that quantum computation is polynomially equivalent to classical probabilistic computation augmented with an oracle for estimating QSWEs.
• It demonstrates that the constrained variant of QSWE estimation can be efficiently solved within the one-bit model of quantum computation.
• Theoretical analysis is used to show that QSWE estimation problems yield problems in BQP that are distinct from previously studied ones.
• The work leverages known results in quantum complexity theory to embed QSWE estimation into the framework of bounded-error quantum polynomial time (BQP).

Experimental results

Research questions

• RQ1 Is quantum computation polynomially equivalent to classical probabilistic computation with an oracle for estimating quadratically signed weight enumerators?
• RQ2 How do the computational powers of unconstrained and constrained QSWE estimation oracles compare to quantum computation?
• RQ3 Can QSWE estimation problems be used to define a canonical BQP-complete problem?
• RQ4 What new complexity classes emerge from the study of QSWE estimation in relation to quantum computation?
• RQ5 How does the one-bit model of quantum computation relate to the solvability of constrained QSWE estimation?

Key findings

• Quantum computation is polynomially equivalent to classical probabilistic computation with an oracle for estimating quadratically signed weight enumerators.
• The unconstrained variant of QSWE estimation may define an oracle more powerful than quantum computation, suggesting potential separation from BQP.
• The constrained variant of QSWE estimation is efficiently solvable in the one-bit model of quantum computation, indicating its limited computational strength.
• QSWE estimation gives rise to a canonical BQP-complete problem, providing a new benchmark for quantum computational hardness.
• The study reveals new complexity classes that are intrinsically linked to quantum computation through QSWE oracles.
• The results deepen the understanding of BQP and its relationship with classical complexity classes by grounding them in algebraic structures tied to error-correcting codes.