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Quantum Threshold is Powerful

Daniel Grier, Jackson Morris

TL;DR

It is proved that Threshold is indeed powerful--there are polynomial-size constant-depth quantum circuits with Threshold gates that compute Fanout to high fidelity, a generalization of a proof by Rosenthal that exponential-size constant-depth circuits with generalized Toffoli gates can compute Fanout.

Abstract

In 2005, Høyer and Špalek showed that constant-depth quantum circuits augmented with multi-qubit Fanout gates are quite powerful, able to compute a wide variety of Boolean functions as well as the quantum Fourier transform. They also asked what other multi-qubit gates could rival Fanout in terms of computational power, and suggested that the quantum Threshold gate might be one such candidate. Threshold is the gate that indicates if the Hamming weight of a classical basis state input is greater than some target value. We prove that Threshold is indeed powerful--there are polynomial-size constant-depth quantum circuits with Threshold gates that compute Fanout to high fidelity. Our proof is a generalization of a proof by Rosenthal that exponential-size constant-depth circuits with generalized Toffoli gates can compute Fanout. Our construction reveals that other quantum gates able to "weakly approximate" Parity can also be used as substitutes for Fanout.

Quantum Threshold is Powerful

TL;DR

It is proved that Threshold is indeed powerful--there are polynomial-size constant-depth quantum circuits with Threshold gates that compute Fanout to high fidelity, a generalization of a proof by Rosenthal that exponential-size constant-depth circuits with generalized Toffoli gates can compute Fanout.

Abstract

In 2005, Høyer and Špalek showed that constant-depth quantum circuits augmented with multi-qubit Fanout gates are quite powerful, able to compute a wide variety of Boolean functions as well as the quantum Fourier transform. They also asked what other multi-qubit gates could rival Fanout in terms of computational power, and suggested that the quantum Threshold gate might be one such candidate. Threshold is the gate that indicates if the Hamming weight of a classical basis state input is greater than some target value. We prove that Threshold is indeed powerful--there are polynomial-size constant-depth quantum circuits with Threshold gates that compute Fanout to high fidelity. Our proof is a generalization of a proof by Rosenthal that exponential-size constant-depth circuits with generalized Toffoli gates can compute Fanout. Our construction reveals that other quantum gates able to "weakly approximate" Parity can also be used as substitutes for Fanout.

Paper Structure

This paper contains 16 sections, 16 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Comparison to the classical setting
  3. Proof techniques and overview
  4. Related work
  5. Future directions
  6. Preliminaries
  7. Multiqubit Gates
  8. Quantum Circuit Complexity Classes
  9. Approximate Quantum Circuits
  10. Bootstrapping weak parity gates
  11. Grid Construction
  12. Removing the Toffoli gates
  13. Quantum MOD gates are powerful even on their own
  14. Simulating qudit arithmetic in QNC[̂p]
  15. Acknowledgements
  16. ...and 1 more sections

Key Result

Theorem 1

There are poly-size constant-depth quantum circuits consisting of Threshold gates and arbitrary single-qubit gates that compute Fanout with high fidelity. Formally, $\ComplexityFont{BQTC}^0 = \BQNC^0_{wf}$.

Figures (2)

  • Figure 1: Constructing a nekomata from $U_S$ and Toffoli gates. Target qubits are shown in blue.
  • Figure 2: Computing $\mathsf P_n$ with a circuit $C$ which prepares an exact $n$-nekomata and its inverse $C^{\dag}$

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 28 more