[Paper Review] Quantum Computation with Harmonic Oscillators

Stephen D. Bartlett, Barry C. Sanders+2 more|ArXiv.org|Nov 20, 2000

Quantum Mechanics and Applications4 citations

TL;DR

This paper proposes a novel continuous-variable quantum computation framework using harmonic oscillators, encoding qudits in number and phase states via the Pegg-Barnett phase operator. It demonstrates that a SUM gate—essential for universality—can be implemented using a Kerr nonlinearity in a microwave cavity, enabling universal quantum computation with well-defined, finite-d qudit states without requiring infinite squeezing.

ABSTRACT

By encoding a qudit in a harmonic oscillator and investigating the infinite limit, we give an entirely new realization of continuous-variable quantum computation. The generalized Pauli group is generated by number and phase operators for harmonic oscillators. We describe a physical realization in terms of modes in a microwave cavity, coupled via a standard Kerr nonlinearity.

Motivation & Objective

• To develop a new continuous-variable quantum computation model based on harmonic oscillators, avoiding reliance on position eigenstates.
• To address the challenge of implementing universal quantum computation in CV systems with well-defined, finite-d computational bases.
• To provide a physically realizable implementation using microwave cavity modes and atomic interactions.
• To demonstrate that the SUM gate can be realized via a standard Kerr nonlinearity, enabling universal quantum computation.
• To show that number and phase states form a dual, well-defined computational basis for qudits in the d→∞ limit, avoiding infinite-squeezing requirements.

Proposed method

• Encode qudits in the number state basis |n⟩ or phase state basis |ϕ⟩, where n, ϕ ∈ {0, 1, ..., d−1}, using the harmonic oscillator Hilbert space.
• Define the generalized Pauli group via the number operator ŜN and the Pegg-Barnett phase operator Ŝθ, which generate unitary transformations.
• Implement single-qudit operations using displacement, squeezing (via χ(2) nonlinearity), and Kerr nonlinearities (via χ(3) nonlinearity) in a microwave cavity.
• Realize two-qudit entangling gates via the four-wave mixing Hamiltonian H = χŜN₁ŜN₂, which generates the SUM gate when time t = χ⁻¹.
• Use single-atom interactions in a micromaser cavity to conditionally prepare states, perform unitary operations, and measure photon number via QND measurements.
• Perform von Neumann measurements in the number basis using momentum-dependent atom-field coupling, where atomic momentum distribution reveals photon number.

Experimental results

Research questions

• RQ1 Can a universal continuous-variable quantum computation be achieved using number and phase states of harmonic oscillators instead of position eigenstates?
• RQ2 How can the SUM gate be physically realized in a CV system using standard nonlinear optical elements?
• RQ3 What is the role of the Pegg-Barnett phase operator in constructing a well-defined, finite-d computational basis for CV quantum computation?
• RQ4 Can microwave cavity systems with atomic interactions implement all required operations (state preparation, unitary evolution, entangling gates, measurement) for universal quantum computation?
• RQ5 How does the d→∞ limit of qudit computation relate to continuous-variable quantum information, and what advantages does it offer over position-eigenstate-based CV models?

Key findings

• The generalized Pauli group generated by number and Pegg-Barnett phase operators provides a well-defined, finite-d structure for CV quantum computation.
• The SUM gate is realized via a Kerr nonlinearity with Hamiltonian χŜN₁ŜN₂, producing the transformation |s₁⟩₁⊗|s₂⟩₂ ↦ |s₁⟩₁⊗|χts₁ + s₂⟩₂ at t = χ⁻¹.
• Number and phase states form a dual basis with ⟨ϕ|n⟩ = exp(iϕn), enabling unitary transformations between computational bases.
• Universal quantum computation is achievable using only displacement, squeezing, and Kerr nonlinearities, which can approximate any polynomial Hamiltonian in â and â†.
• State preparation, unitary operations, and measurement can be implemented deterministically using atomic sequences in a high-Q microwave cavity with controlled interactions.
• The proposed approach avoids the need for infinite-squeezing of Gaussian states, as computational basis states are finite-d and physically preparable with high fidelity.