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Quantum Computational Complexity

John Watrous

TL;DR

This survey maps the landscape of quantum computational complexity, focusing on three core notions: efficient polynomial-time quantum computation ($BQP$), efficient verification with quantum proofs ($QMA$), and quantum interactive proofs ($QIP$). It builds the framework from the quantum circuit model to complexity-class definitions, analyzes error reduction and oracle results, and surveys quantum proofs, interactive proofs, and other notions such as quantum advice and space-bounded models. Key contributions include the BQP subroutine theorem, containment and separation results with classical classes, and the exploration of zero-knowledge and multi-prover quantum proofs, highlighting both established results and major open questions. The discussion clarifies how quantum mechanics informs computational hardness, outlines robust relationships between quantum and classical complexity, and points to rich avenues for future research, including the power of entangled multi-prover systems and the status of prominent problems like graph isomorphism within quantum complexity.

Abstract

This article surveys quantum computational complexity, with a focus on three fundamental notions: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems. Properties of quantum complexity classes based on these notions, such as BQP, QMA, and QIP, are presented. Other topics in quantum complexity, including quantum advice, space-bounded quantum computation, and bounded-depth quantum circuits, are also discussed.

Quantum Computational Complexity

TL;DR

This survey maps the landscape of quantum computational complexity, focusing on three core notions: efficient polynomial-time quantum computation (), efficient verification with quantum proofs (), and quantum interactive proofs (). It builds the framework from the quantum circuit model to complexity-class definitions, analyzes error reduction and oracle results, and surveys quantum proofs, interactive proofs, and other notions such as quantum advice and space-bounded models. Key contributions include the BQP subroutine theorem, containment and separation results with classical classes, and the exploration of zero-knowledge and multi-prover quantum proofs, highlighting both established results and major open questions. The discussion clarifies how quantum mechanics informs computational hardness, outlines robust relationships between quantum and classical complexity, and points to rich avenues for future research, including the power of entangled multi-prover systems and the status of prominent problems like graph isomorphism within quantum complexity.

Abstract

This article surveys quantum computational complexity, with a focus on three fundamental notions: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems. Properties of quantum complexity classes based on these notions, such as BQP, QMA, and QIP, are presented. Other topics in quantum complexity, including quantum advice, space-bounded quantum computation, and bounded-depth quantum circuits, are also discussed.

Paper Structure

This paper contains 34 sections, 17 theorems, 22 equations, 14 figures.

Table of Contents

  1. Definition of the subject and its importance
  2. Introduction
  3. Computational complexity
  4. Quantum information
  5. The quantum circuit model
  6. General quantum circuits
  7. A finite universal gate set
  8. Unitary purifications of quantum circuits
  9. Oracles in the quantum circuit model
  10. Polynomial-time quantum computations
  11. Polynomial-time generated circuit families and BQP
  12. Error reduction for BQP
  13. Simulating classical computations with quantum circuits
  14. The BQP subroutine theorem
  15. Classical upper bounds on BQP
  16. ...and 19 more sections

Key Result

Theorem 1

Let $\Phi$ be an arbitrary quantum operation from $n$ qubits to $m$ qubits. Then for every $\varepsilon>0$ there exists a quantum circuit $Q$ with $n$ input qubits and $m$ output qubits such that $\delta(\Phi,Q) < \varepsilon$. Moreover, for fixed $n$ and $m$, the circuit $Q$ may be taken to satisfy

Figures (14)

  • Figure 1: A diagram illustrating known inclusions among most of the classical complexity classes discussed in this paper. Lines indicate containments going upward; for example, AM is contained in PSPACE.
  • Figure 2: An example of a quantum circuit. The input qubits are labelled $\mathsf{X}_1,\ldots,\mathsf{X}_4$, the output qubits are labelled $\mathsf{Y}_1,\ldots,\mathsf{Y}_3$, and the gates are labelled by (hypothetical) quantum operations $\Phi_1,\ldots,\Phi_6$.
  • Figure 3: A universal collection of quantum gates: Toffoli, Hadamard, phase-shift, ancillary, and erasure gates.
  • Figure 4: Four additional quantum gates, together with their implementations as quantum circuits. Top left: a NOT gate. Top right: a constant $\mathinner{|{1}\rangle}$ ancillary gate. Bottom left: a controlled-NOT gate. Bottom right: a phase-damping (or decoherence) gate.
  • Figure 5: A general quantum circuit (left) and its unitary purification (right).
  • ...and 9 more figures

Theorems & Definitions (18)

  • Theorem 1: Universality Theorem
  • Definition 2
  • Proposition 3: Error reduction for BQP
  • Theorem 4: BQP subroutine theorem
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 8 more