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Quantum Supremacy through the Quantum Approximate Optimization Algorithm

Edward Farhi, Aram W Harrow

TL;DR

This work analyzes the potential of the Quantum Approximate Optimization Algorithm ($QAOA$) to achieve quantum supremacy on near-term quantum devices by showing that sampling from its output distribution would imply a collapse of the Polynomial Hierarchy ($PH$) under standard complexity assumptions. It establishes a deep connection between $QAOA$ and the broader class of postselected computations: the authors prove $PostQAOA = PostBQP$, and they further show that efficient classical sampling of $QAOA$ outputs would collapse $PH$, thereby arguing for the inherent hardness of classical simulation. The paper also contrasts $QAOA$ with the Quantum Adiabatic Algorithm ($QADI$) under stoquastic, gapped conditions, arguing that stoquastic $QADI$ is comparatively easier to simulate classically via Path-Integral Monte Carlo and that universal adiabatic computing requires nonstoquasticity. Overall, the authors advocate pursuing $QAOA$ on near-term devices as a path to both practical optimization and a robust demonstration of quantum supremacy, while highlighting the subtleties and limitations of simulating stoquastic $QADI$.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is designed to run on a gate model quantum computer and has shallow depth. It takes as input a combinatorial optimization problem and outputs a string that satisfies a high fraction of the maximum number of clauses that can be satisfied. For certain problems the lowest depth version of the QAOA has provable performance guarantees although there exist classical algorithms that have better guarantees. Here we argue that beyond its possible computational value the QAOA can exhibit a form of Quantum Supremacy in that, based on reasonable complexity theoretic assumptions, the output distribution of even the lowest depth version cannot be efficiently simulated on any classical device. We contrast this with the case of sampling from the output of a quantum computer running the Quantum Adiabatic Algorithm (QADI) with the restriction that the Hamiltonian that governs the evolution is gapped and stoquastic. Here we show that there is an oracle that would allow sampling from the QADI but even with this oracle, if one could efficiently classically sample from the output of the QAOA, the Polynomial Hierarchy would collapse. This suggests that the QAOA is an excellent candidate to run on near term quantum computers not only because it may be of use for optimization but also because of its potential as a route to establishing quantum supremacy.

Quantum Supremacy through the Quantum Approximate Optimization Algorithm

TL;DR

This work analyzes the potential of the Quantum Approximate Optimization Algorithm () to achieve quantum supremacy on near-term quantum devices by showing that sampling from its output distribution would imply a collapse of the Polynomial Hierarchy () under standard complexity assumptions. It establishes a deep connection between and the broader class of postselected computations: the authors prove , and they further show that efficient classical sampling of outputs would collapse , thereby arguing for the inherent hardness of classical simulation. The paper also contrasts with the Quantum Adiabatic Algorithm () under stoquastic, gapped conditions, arguing that stoquastic is comparatively easier to simulate classically via Path-Integral Monte Carlo and that universal adiabatic computing requires nonstoquasticity. Overall, the authors advocate pursuing on near-term devices as a path to both practical optimization and a robust demonstration of quantum supremacy, while highlighting the subtleties and limitations of simulating stoquastic .

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is designed to run on a gate model quantum computer and has shallow depth. It takes as input a combinatorial optimization problem and outputs a string that satisfies a high fraction of the maximum number of clauses that can be satisfied. For certain problems the lowest depth version of the QAOA has provable performance guarantees although there exist classical algorithms that have better guarantees. Here we argue that beyond its possible computational value the QAOA can exhibit a form of Quantum Supremacy in that, based on reasonable complexity theoretic assumptions, the output distribution of even the lowest depth version cannot be efficiently simulated on any classical device. We contrast this with the case of sampling from the output of a quantum computer running the Quantum Adiabatic Algorithm (QADI) with the restriction that the Hamiltonian that governs the evolution is gapped and stoquastic. Here we show that there is an oracle that would allow sampling from the QADI but even with this oracle, if one could efficiently classically sample from the output of the QAOA, the Polynomial Hierarchy would collapse. This suggests that the QAOA is an excellent candidate to run on near term quantum computers not only because it may be of use for optimization but also because of its potential as a route to establishing quantum supremacy.

Paper Structure

This paper contains 17 sections, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Constraint satisfaction problems
  4. Quantum Approximate Optimization Algorithm (QAOA)
  5. The Polynomial Hierarchy
  6. Counting
  7. Efficient Classical Computation of Matrix Elements of Quantum Circuits Implies that $\mathop{\mathrm{\mathsf{P}}}\nolimits=\mathop{\mathrm{\mathsf{NP}}}\nolimits$
  8. Post-Selected Quantum Computing
  9. Efficient Classical Sampling of the Output of an Arbitrary Quantum Circuit Implies the Collapse of the Polynomial Hierarchy
  10. $\mathop{\mathrm{\mathsf{PostQAOA}}}\nolimits = \mathop{\mathrm{\mathsf{PostBQP}}}\nolimits$
  11. Efficient Classical Sampling of the Output of the $p=1$ QAOA Implies the Collapse of the Polynomial Hierarchy
  12. Quantum Adiabatic Algorithm
  13. Definition and background
  14. Classical simulation of the Quantum Adiabatic Algorithm
  15. Formal Evidence that Stoquastic Gapped Adiabatic Evolution is Easier to Simulate than QAOA
  16. ...and 2 more sections

Figures (1)

  • Figure 1: (a) Somewhere inside of a big quantum circuit consisting of Hadamards and diagonal unitaries, a Hadamard acts on qubit $j$. The quantum state before the Hadamard acts is $|\alpha\rangle$ which we denote as $|\alpha\rangle_{j, \text{rest}}$ to keep track of how it can be decomposed into qubit $j$ and the rest. This circuit element can be replaced by the post-selected circuit in (b). Qubit $j$ has been replaced by an auxiliary qubit and the new circuit is of the form (\ref{['eq:tilde-QAOA']}) with qubit $j$ post-selected to be $|0\rangle$.