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Adaptive Quantum Computation, Constant Depth Quantum Circuits and Arthur-Merlin Games

Barbara M. Terhal, David P. DiVincenzo

TL;DR

The paper investigates whether constant-depth quantum circuits with only two-qubit gates can realize computations that resist efficient classical simulation. By formulating adaptive and nonadaptive (fixed-guess) quantum circuit models and defining density-computation and sampling-based simulators, it links classical simulability to major complexity-class consequences. The main contributions show that (i) an efficient density-computation simulation for these constant-depth circuits would force ${BPP}={BQP}$ and collapse the polynomial hierarchy, and (ii) even with Merlin guiding the simulation, ${BQP}\subseteq AM$ would hold; additionally, the Gottesman-Chuang construction is recast into a constant-depth nonadaptive form (depth at most four), highlighting nontrivial quantum computation at restricted depth. These results connect circuit depth to foundational complexity-theoretic questions and suggest that constant-depth, teleportation-based quantum models may exhibit nonclassical power with potential experimental relevance in linear-optics implementations like KLM.

Abstract

We present evidence that there exist quantum computations that can be carried out in constant depth, using 2-qubit gates, that cannot be simulated classically with high accuracy. We prove that if one can simulate these circuits classically efficiently then the complexity class BQP is contained in AM.

Adaptive Quantum Computation, Constant Depth Quantum Circuits and Arthur-Merlin Games

TL;DR

The paper investigates whether constant-depth quantum circuits with only two-qubit gates can realize computations that resist efficient classical simulation. By formulating adaptive and nonadaptive (fixed-guess) quantum circuit models and defining density-computation and sampling-based simulators, it links classical simulability to major complexity-class consequences. The main contributions show that (i) an efficient density-computation simulation for these constant-depth circuits would force and collapse the polynomial hierarchy, and (ii) even with Merlin guiding the simulation, would hold; additionally, the Gottesman-Chuang construction is recast into a constant-depth nonadaptive form (depth at most four), highlighting nontrivial quantum computation at restricted depth. These results connect circuit depth to foundational complexity-theoretic questions and suggest that constant-depth, teleportation-based quantum models may exhibit nonclassical power with potential experimental relevance in linear-optics implementations like KLM.

Abstract

We present evidence that there exist quantum computations that can be carried out in constant depth, using 2-qubit gates, that cannot be simulated classically with high accuracy. We prove that if one can simulate these circuits classically efficiently then the complexity class BQP is contained in AM.

Paper Structure

This paper contains 11 sections, 8 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous work
  3. Definitions
  4. Basic Concepts
  5. Adaptive and nonadaptive quantum computation
  6. Classical Simulations
  7. Constant-Depth Quantum Circuits
  8. General Simulations
  9. Conclusion
  10. Appendix: Bell outcomes
  11. Appendix: Approximate Set Size

Key Result

Proposition 1

If an efficient density computation exists, it provides the means for performing an efficient simulation with $\epsilon=0$.

Figures (1)

  • Figure 1: (a) The Gottesman-Chuang implementation of the CNOT gate by teleportation. In addition to the two qubit inputs $q_1$ and $q_2$, the teleportation circuit has four additional ancilla qubit inputs ("1", "2", "3", "4") preset to the entangled state $| \Psi_C \rangle$. Two Bell measurements (B.M.) are performed as indicated, resulting in two bit-pairs ${\bf b}_1$ and ${\bf b}_2$ as output. These bit pairs determine the parameters of two one-qubit quantum gates $U$ and $V$. (b) Construction of the entangled state $| \Psi_C \rangle$. $H$ is the one-qubit Hadamard gate, specified by the $2\times 2$ matrix ${1\,\,1\choose 1 -1}/\sqrt{2}$. The dotted box (which can be completed in one time step) causes the creation of the entangled state $\Phi^+$ between ancilla bits "1" and "2" and between "3" and "4". The final CNOT constitutes the "offline" application of the two-qubit gate as mentioned in the text.

Theorems & Definitions (26)

  • Definition 1: Quantum Register $QR(w)$
  • Definition 2: Quantum Gate
  • Definition 3: Quantum Measurement $\mathcal{M}$ in the Standard Basis
  • Definition 4: Quantum Circuit $QC_x(w,w',| init \rangle,d)$
  • Definition 5: BQP
  • Definition 6: AM
  • Definition 7: ${\cal QC}_{ad}$
  • Definition 8: ${\cal QC}_{nad}$
  • Definition 9: $S_\epsilon({\cal QC})$
  • Definition 10: $S^C({\cal QC})$
  • ...and 16 more