[Paper Review] Nonunitary quantum circuit

Hiroaki Terashima, Masahito Ueda|arXiv (Cornell University)|Apr 9, 2003

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper proposes a nonunitary quantum circuit framework where quantum gates are implemented via quantum measurements, enabling probabilistic but universal quantum computation without ancilla qubits. It demonstrates that controlled-NOT and single-qubit unitary gates form a universal set for such circuits, and introduces a reversing measurement scheme to boost success probabilities, offering a path to fault-tolerant quantum computation with reduced qubit overhead.

ABSTRACT

A quantum circuit is generalized to a nonunitary one whose constituents are nonunitary gates operated by quantum measurement. It is shown that a specific type of one-qubit nonunitary gates, the controlled-NOT gate, as well as all one-qubit unitary gates constitute a universal set of gates for the nonunitary quantum circuit, without the necessity of introducing ancilla qubits. A reversing measurement scheme is used to improve the probability of successful nonunitary gate operation. A quantum NAND gate and Abrams-Lloyd's nonlinear gate are analyzed as examples. Our nonunitary circuit can be used to reduce the qubit overhead needed to ensure fault-tolerant quantum computation.

Motivation & Objective

• To develop a general framework for quantum computation based on nonunitary operations, extending beyond unitary circuits.
• To address the high qubit overhead in fault-tolerant quantum computation by enabling probabilistic, nonunitary gate operations.
• To show that nonunitary gates, implemented via quantum measurement, can form a universal set without requiring ancilla qubits.
• To improve the success probability of nonunitary gate operations using a reversing measurement scheme.
• To explore the implications of nonunitary operations for solving NP-complete problems in polynomial time, if unit-probability implementation were possible.

Proposed method

• Defines a nonunitary gate operation as |ψ⟩ → N|ψ⟩ / √⟨ψ|N†N|ψ⟩, where N is a nonunitary operator acting on the state.
• Uses quantum measurement to physically implement nonunitary gates, making the logical operation nonunitary at both physical and logical levels.
• Applies a reversing measurement scheme to increase the probability of successful gate operation by post-selecting favorable measurement outcomes.
• Demonstrates universality by showing that the controlled-NOT gate and single-qubit unitaries form a universal gate set for nonunitary circuits.
• Analyzes specific examples, including a quantum NAND gate and Abrams-Lloyd’s nonlinear gate, to illustrate the framework’s applicability.
• Quantifies the exponential decay of success probability with circuit depth, showing it scales as (1/6)^n for Abrams-Lloyd’s gate.

Experimental results

Research questions

• RQ1 Can a universal set of nonunitary quantum gates be constructed without introducing ancilla qubits?
• RQ2 How can the success probability of nonunitary gate operations be maximized given their probabilistic nature?
• RQ3 What is the relationship between nonunitary quantum computation and the solution of NP-complete problems?
• RQ4 To what extent can nonunitary operations reduce qubit overhead in fault-tolerant quantum computation?
• RQ5 Can nonunitary gates be optimized using measurement-based feedback, such as reversing measurements?

Key findings

• A universal set of nonunitary quantum gates is formed by the controlled-NOT gate, single-qubit unitary gates, and a specific class of nonunitary one-qubit gates, without requiring ancilla qubits.
• The reversing measurement scheme increases the success probability of nonunitary gate operations to the maximum allowed by the measurement process.
• The success probability of a nonunitary quantum circuit decays exponentially with the number of nonunitary gates, scaling as (1/6)^n for Abrams-Lloyd’s nonlinear gate.
• If nonunitary gates could be applied with unit probability, NP-complete problems could be solved in polynomial time, paralleling Abrams-Lloyd’s result in nonlinear quantum theory.
• Nonunitary quantum circuits can implement the quantum NAND gate and Abrams-Lloyd’s nonlinear gate, demonstrating practical utility in reducing qubit requirements.
• The framework establishes a direct link between physical nonunitarity and computational power, suggesting that nonunitary operations could enable new classes of efficient quantum algorithms.