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Quantum Electronics and Laser Science Conference

Quantum computation with harmonic oscillators

Stephen D. Bartlett, Barry C. Sanders, Hubert de Guise

Published Nov 13, 2002• 3 citations
Publisher

Abstract

Summary form only given. The use of continuous-variable (CV) quantum computing allows information to be encoded and processed much more compactly and efficiently than with discrete-variable (qubit) computing. That is, with CV realizations, one can perform quantum information processes using fewer coupled quantum systems: a considerable advantage for the experimental realization of quantum computing. As a result, CV quantum information theory is of interest in quantum error correction, quantum cryptography and quantum teleportation. For example, quantum teleportation has recently been tested experimentally in the CV domain. The current proposed realization of CV quantum computation employs position eigenstates as a computational basis; these states are approximated in experiment using highly squeezed states. We introduce new CV representations, where the generalized Pauli group is generated by the number and phase operators for harmonic oscillators, and the computational basis is given by either harmonic oscillator energy eigenstates or phase eigenstates.

DOI
10.1109/qels.2001.962046

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Summary form only given. The use of continuous-variable (CV) quantum computing allows information to be encoded and processed much more compactly and efficiently than with discrete-variable (qubit) computing. That is, with CV realizations, one can perform quantum information processes using fewer coupled quantum systems: a considerable advantage for the experimental realization of quantum computing. As a result, CV quantum information theory is of interest in quantum error correction, quantum cryptography and quantum teleportation. For example, quantum teleportation has recently been tested experimentally in the CV domain. The current proposed realization of CV quantum computation employs position eigenstates as a computational basis; these states are approximated in experiment using highly squeezed states. We introduce new CV representations, where the generalized Pauli group is generated by the number and phase operators for harmonic oscillators, and the computational basis is given by either harmonic oscillator energy eigenstates or phase eigenstates.
Summary form only given. We are working on hybrid quantum information processing, which combines two methodologies of quantum information processing - qubit and continuous variable (CV) [1]. More precisely, we encode logical qubits by using CV methodology and utilize CV quantum processors for the realization of a fault-tolerant large-scale universal optical quantum computer. The advantage of this methodology is that we can have both high-fidelity nature of qubits and determinisity of CV quantum processors. In other words, we can enjoy both particle- and wave-nature of quantum mechanics. Towards this goal we performed various things, which include quantum error correction with nine-party CV entanglement [2], teleportation of Schrödinger's cat state [3], adaptive homodyne measurement with phase-squeezed states [4], deterministic teleportation of time-bin qubits [5], creation of ultra-large-scale CV cluster states [6], generation and measurement of CV entanglement on a chip [7], and synchronization of photons with cavity-based quantum memories [8].
Abstract The stable operation of quantum computers will rely on error correction, in which single quantum bits of information are stored redundantly in the Hilbert space of a larger system. Such encoded qubits are commonly based on arrays of many physical qubits, but can also be realized using a single higher-dimensional quantum system, such as a harmonic oscillator1–3. In such a system, a powerful encoding has been devised based on periodically spaced superpositions of position eigenstates4–6. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach7–11. Here we demonstrate such an encoded qubit using a superposition of displaced squeezed states of the harmonic motion of a single trapped 40Ca+ ion, controlling and measuring the mechanical oscillator through coupling to an ancillary internal-state qubit12. We prepare and reconstruct logical states with an average squared fidelity of 87.3 ± 0.7 per cent. Also, we demonstrate a universal logical single-qubit gate set, which we analyse using process tomography. For Pauli gates we reach process fidelities of about 97 per cent, whereas for continuous rotations we use gate teleportation and achieve fidelities of approximately 89 per cent. This control method opens a route for exploring continuous variable error correction as well as hybrid quantum information schemes using both discrete and continuous variables13. The code states also have direct applications in quantum sensing, allowing simultaneous measurement of small displacements in both position and momentum14, 15.
Quantum information can be encoded in a scalable manner over the continuous variables (CV) that are the canonically conjugate quantum amplitudes of the electromagnetic field, which are mathematical equivalents of the position and momentum of the quantum harmonic oscillator. Previous results in our group in Charlottesville and the groups of Akira Furusawa in Tokyo, of Nicolas Treps in Paris, and of Ping-Koy Lam in Canberra, have shown that CV quantum information is massively scalable. In addition, Nicolas Menicucci in Melbourne has shown that there exists a fault tolerance threshold for CV quantum computing with the Gottesman-Kitaev-Preskill qubit error encoding. Here, we report on our exploration of the extension of scalable CV encoding by way of intramode, rather than intermode, squeezing and entanglement. As is well known, single-mode squeezing can only be considered along with the "quasi-mode" approximation which assimilates the modes of the optical parametric oscillator cavity used in the experiment to delta functions. As Hans Bachor and Tim Ralph noted, when one ambles beyond this approximation, single-mode squeezing can be described as intramode two-mode squeezing. Here, we aim at generalizing this situation to the case of multipartite entangled cluster states.
We introduce a framework for simulating hybrid oscillator-qubit quantum processors on qubit-only systems through position encoding. By encoding continuous-variable position and momentum wave functions into qubit amplitudes, our method efficiently simulates all Gaussian and conditional Gaussian operations -- encompassing the phase-space instruction set (beam splitter, single-qubit rotation, conditional displacement) and extending to squeezing, conditional squeezing, conditional rotation, and conditional beam splitter -- using O ⁣(log⁡2 ⁣(Γ+log⁡(1/ε)))O\!\left(\log^2\!\left(Γ+ \log(1/ε)\right)\right)O(log2(Γ+log(1/ε))) qubit gates per hybrid gate, where ΓΓΓ is the Fock-level bound and εεε is the target precision. This polylogarithmic per-gate complexity represents an exponential improvement over Fock basis encoding approaches, which require exponential quantum or classical resources in the number of qubits per mode. We provide rigorous numerical characterization of quantum Fourier transform errors for Fock-bounded states, enabling precise resource estimation for practical implementations. This work establishes that hybrid oscillator-qubit algorithms can be implemented on qubit processors with polynomial overhead, providing new insights into the computational power trade-offs between discrete-variable and hybrid continuous-discrete-variable quantum computing.