[Paper Review] On the Complexity of Random Quantum Computations and the Jones Polynomial

Ryan L. Mann, Michael J. Bremner|arXiv (Cornell University)|Nov 2, 2017

Quantum Computing Algorithms and Architecture2 references9 citations

TL;DR

This paper establishes a direct link between the classical complexity of simulating random quantum computations and the average-case hardness of approximating Jones polynomials. Under the assumptions that the Polynomial Hierarchy does not collapse and that relative-error approximations of Jones polynomials are as hard on average as in the worst case, the authors prove that classical simulation of random quantum circuits up to constant total variation distance is impossible.

ABSTRACT

There is a natural relationship between Jones polynomials and quantum computation. We use this relationship to show that the complexity of evaluating relative-error approximations of Jones polynomials can be used to bound the classical complexity of approximately simulating random quantum computations. We prove that random quantum computations cannot be classically simulated up to a constant total variation distance, under the assumption that (1) the Polynomial Hierarchy does not collapse and (2) the average-case complexity of relative-error approximations of the Jones polynomial matches the worst-case complexity over a constant fraction of random links. Our results provide a straightforward relationship between the approximation of Jones polynomials and the complexity of random quantum computations.

Motivation & Objective

• To establish a connection between the classical complexity of simulating random quantum computations and the average-case complexity of approximating Jones polynomials.
• To investigate whether relative-error approximations of Jones polynomials are as hard on average as in the worst case, a key assumption for complexity bounds.
• To extend existing quantum advantage arguments—previously based on permanents and other hard functions—to the Jones polynomial as a new candidate for quantum computational supremacy.
• To provide a formal proof that classical simulation of random quantum circuits up to constant total variation distance is impossible, assuming standard complexity-theoretic conjectures.
• To leverage approximate unitary designs and anti-concentration bounds to strengthen the complexity-theoretic argument for quantum advantage using Jones polynomials.

Proposed method

• Uses the known quantum-computational universality of the Jones polynomial to relate its approximation complexity to quantum circuit sampling problems.
• Applies Stockmeyer's Counting Theorem to approximate the Jones polynomial within a relative error using access to NP and FBPP classes.
• Employs approximate unitary t-designs (for t ≥ 2) to model random quantum circuits and derive anti-concentration bounds on matrix elements.
• Uses the Paley-Zygmund inequality to bound the probability that matrix elements of random unitaries exceed a threshold, relying on approximate design conditions.
• Combines anti-concentration bounds with approximate counting in the Polynomial Hierarchy to show that classical simulation would imply PH collapse.
• Relies on the conjecture that average-case complexity of relative-error Jones polynomial approximations matches worst-case complexity over a constant fraction of random links.

Experimental results

Research questions

• RQ1 Can the classical complexity of simulating random quantum circuits be bounded using the average-case hardness of approximating Jones polynomials?
• RQ2 Under what conditions does the average-case complexity of relative-error approximations of the Jones polynomial match its worst-case complexity?
• RQ3 Does the hardness of approximating the Jones polynomial imply that random quantum circuits cannot be classically simulated up to constant total variation distance?
• RQ4 How do approximate unitary designs and anti-concentration properties of matrix elements support the quantum advantage argument based on Jones polynomials?
• RQ5 What is the role of the Polynomial Hierarchy in ruling out efficient classical simulation of random quantum circuits via Jones polynomial approximation?

Key findings

• Under the assumption that the Polynomial Hierarchy does not collapse, classical simulation of random quantum circuits up to constant total variation distance is impossible.
• The paper proves that if relative-error approximations of the Jones polynomial are as hard on average as in the worst case (Conjecture 1), then classical simulation of random quantum circuits is classically intractable.
• The authors establish that unitary matrices from an ε-approximate t-design (t ≥ 2) satisfy anti-concentration bounds: the probability that |⟨α|U|β⟩|² exceeds γ/d is at least (1−ε−γ)²/(2(1+ε)) for 0 ≤ γ ≤ 1−ε.
• Using Stockmeyer’s Counting Theorem, the paper shows that a relative-error approximation to the Jones polynomial can be achieved in FBPP^NP^C, enabling classical approximation within poly(n) relative error.
• The proof relies on bounding the l1 distance between the output distribution of a random quantum circuit and the ideal distribution via anti-concentration and approximate counting.
• The result extends the framework of quantum supremacy arguments beyond permanents and other counting problems to the Jones polynomial, broadening the class of functions that can serve as evidence for quantum advantage.