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Results on two-bit gate design for quantum computers

David P. DiVincenzo, John Smolin

TL;DR

Problem: determine finite two‑qubit gate sequences that realize arbitrary three‑qubit unitaries and quantify minimal gate counts for key operations like Toffoli. Approach: brute‑force numerical search over gate topologies with a nonlinear optimization to match target unitaries exactly. Findings: any $U(8)$ can be realized with six two‑qubit gates; Toffoli requires five; phase sensitivity can reduce to three in special topologies (Margolus) but simple phase changes often require more gates, underscoring the cost of precise phase control. Implications: provides concrete gate‑count benchmarks and emphasizes phase accuracy as a critical constraint in practical quantum gate synthesis.

Abstract

We present numerical results which show how two-bit logic gates can be used in the design of a quantum computer. We show that the Toffoli gate, which is a universal gate for all classical reversible computation, can be implemented using a particular sequence of exactly five two-bit gates. An arbitrary three-bit unitary gate, which can be used to build up any arbitrary quantum computation, can be implemented exactly with six two-bit gates. The ease of implementation of any particular quantum operation is dependent upon a very non-classical feature of the operation, its exact quantum phase factor.

Results on two-bit gate design for quantum computers

TL;DR

Problem: determine finite two‑qubit gate sequences that realize arbitrary three‑qubit unitaries and quantify minimal gate counts for key operations like Toffoli. Approach: brute‑force numerical search over gate topologies with a nonlinear optimization to match target unitaries exactly. Findings: any can be realized with six two‑qubit gates; Toffoli requires five; phase sensitivity can reduce to three in special topologies (Margolus) but simple phase changes often require more gates, underscoring the cost of precise phase control. Implications: provides concrete gate‑count benchmarks and emphasizes phase accuracy as a critical constraint in practical quantum gate synthesis.

Abstract

We present numerical results which show how two-bit logic gates can be used in the design of a quantum computer. We show that the Toffoli gate, which is a universal gate for all classical reversible computation, can be implemented using a particular sequence of exactly five two-bit gates. An arbitrary three-bit unitary gate, which can be used to build up any arbitrary quantum computation, can be implemented exactly with six two-bit gates. The ease of implementation of any particular quantum operation is dependent upon a very non-classical feature of the operation, its exact quantum phase factor.

Paper Structure

This paper contains 8 sections, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Simulations
  3. Notation and Constraints
  4. Numerical Technique
  5. Results: arbitrary U(8)
  6. Results: Toffoli gate
  7. Results: Margolus' modification of Toffoli--the phase problem
  8. Concluding Remarks

Figures (2)

  • Figure 1: Illustration of a replacement of a three-bit gate by a sequence of one- and two-bit gates. The numbering convention (1,2,3) for three bits is indicated. The one bit "N" gate may be absorbed into the definition of the two-bit gates. According to the notation discussed in the text, the network shown has the topology (1212). For more details of the use of this particular decomposition, see Divi.
  • Figure 2: Modification of network topology by swapping. The (131) topology of (a) is changed in (b) by replacing (3) by the set of gates in the dotted box: a (2) and two surrounding swapping gates of the (1) type. The pairs of (1)'s are then merged together in (c). The net result is the replacement (131)$\rightarrow$(121).