[Paper Review] The Computational Complexity of Ball Permutations

Scott Aaronson, Adam Bouland+2 more|arXiv (Cornell University)|Oct 21, 2016

Quantum Computing Algorithms and Architecture18 references4 citations

TL;DR

This paper investigates the computational complexity of quantum and classical models based on permuting distinguishable particles (balls), showing that the quantum ball-permuting model with standard basis input is contained in DQC1—indicating weaker-than-universal power—while certain input states enable BQP universality via encoded computation. A restricted, integrable version becomes postselection-universal, implying computational collapse if efficiently classically simulated.

ABSTRACT

Inspired by connections to two dimensional quantum theory, we define several models of computation based on permuting distinguishable particles (which we call balls), and characterize their computational complexity. In the quantum setting, we find that the computational power of this model depends on the initial input states. More precisely, with a standard basis input state, we show how to approximate the amplitudes of this model within additive error using the model DQC1 (the class of problems solvable with one clean qubit), providing evidence that the model in this case is weaker than universal quantum computing. However, for specific choices of input states, the model is shown to be universal for BQP in an encoded sense. We use representation theory of the symmetric group to partially classify the computational complexity of this model for arbitrary input states. Interestingly, we find some input states which yield a model intermediate between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an integrable scattering problem in 1+1 dimensions. We show it is universal under postselection, if we allow intermediate destructive measurements and specific input states. Therefore, the existence of any classical procedure to sample from the output distribution of this model within multiplicative error implies collapse of polynomial hierarchy to its third level. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP. Moreover, we find a nondeterministic version of this model is NP-complete.

Motivation & Objective

• To characterize the computational complexity of quantum and classical models based on permuting distinguishable particles (balls) in one-dimensional systems.
• To investigate how the choice of initial quantum state affects the computational power of the ball-permuting model.
• To determine whether restricted versions of the model—especially those based on integrable scattering in 1+1 dimensions—achieve universal quantum computation under postselection.
• To define and analyze classical probabilistic and nondeterministic variants of the ball-permuting model, identifying their complexity classes.

Proposed method

• Formalizes a quantum model where two-particle partial swap gates act on permutations of n distinguishable balls, mapping |x,y⟩ → c|x,y⟩ + is|y,x⟩ with c² + s² = 1.
• Uses representation theory of the symmetric group Sₙ to decompose the Hilbert space ℂSₙ into irreducible representations (irreps), analyzing the action of ball-permuting gates on these subspaces.
• Applies the bridge lemma and branching rules to study the unitary group generated by the gates, especially for irreps corresponding to Young diagrams with two rows or columns.
• Introduces a restricted model based on integrable 1+1 dimensional scattering, showing it becomes universal under postselection with specific input states.
• Defines classical analogs: a randomized model (RBall) and a nondeterministic model (NBall), analyzing their complexity relative to L, BPP, and NP.
• Uses trace estimation and average trace computation to relate amplitudes in the quantum model to known complexity classes like DQC1 and BQP.

Experimental results

Research questions

• RQ1 What is the computational power of the ball-permuting quantum model when initialized in the standard basis state |12…n⟩?
• RQ2 Can the ball-permuting model achieve BQP universality for specific initial states, and if so, under what conditions?
• RQ3 What is the complexity of the randomized ball-permuting model (RBall), and how does it relate to L, BPP, and Almost-L?
• RQ4 Does the integrable ball-permuting model become universal under postselection, and what are the consequences for classical simulation?
• RQ5 What is the complexity of the counting version of the ball-permuting model, and does it correspond to #P?

Key findings

• The ball-permuting model with initial state |12…n⟩ is contained in DQC1, implying it is unlikely to be universal for BQP and thus substantially weaker than full quantum computation.
• For certain irreducible representations (e.g., two-row or two-column Young diagrams), the model generates a unitary group as large as possible, suggesting potential for BQP universality.
• With specific input states, the model achieves BQP universality in an encoded sense, as the amplitude estimation problem becomes BQP-complete.
• The restricted, integrable ball-permuting model becomes universal under postselection; thus, efficient classical sampling would collapse the polynomial hierarchy to its third level.
• The randomized ball-permuting model (RBall) satisfies BPL ⊆ RBall ⊆ Almost-L, and two adaptive queries to RBall can simulate Almost-L, though one-query simulation remains open.
• The nondeterministic ball-permuting model (NBall) is NP-complete, and the counting version #Ball is unlikely to be approximable within multiplicative error unless P = NP.