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arXiv:0812.3694 · New Journal of Physics

Limitations on continuous variable quantum algorithms with Fourier transforms

Mark Adcock, Peter Høyer, Barry C. Sanders

Published Oct 21, 2009• 38 citations
arXivHTMLPDF

Abstract

We study quantum algorithms implemented within a single harmonic oscillator, or equivalently within a single mode of the electromagnetic field. Logical states correspond to functions of the canonical position, and the Fourier transform to canonical momentum serves as the analogue of the Hadamard transform for this implementation. This continuous variable version of quantum information processing has widespread appeal because of advanced quantum optics technology that can create, manipulate and read Gaussian states of light. We show that, contrary to a previous claim, this implementation of quantum information processing has limitations due to a position–momentum trade-off of the Fourier transform, analogous to the famous time-bandwidth theorem of signal processing.

DOI
10.1088/1367-2630/11/10/103035

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We study quantum algorithms implemented within a single harmonic oscillator, or equivalently within a single mode of the electromagnetic field. Logical states correspond to functions of the canonical position, and the Fourier transform to canonical momentum serves as the analogue of the Hadamard transform for this implementation. This continuous variable version of quantum information processing has widespread appeal because of advanced quantum optics technology that can create, manipulate and read Gaussian states of light. We show that, contrary to a previous claim, this implementation of quantum information processing has limitations due to a position–momentum trade-off of the Fourier transform, analogous to the famous time-bandwidth theorem of signal processing.
Quantum information processing is appealing due its ability to solve certain problems quantitatively faster than classical information processing. Most quantum algorithms have been studied in discretely parameterized systems, but many quantum systems are continuously parameterized. The field of quantum optics in particular has sophisticated techniques for manipulating continuously parameterized quantum states of light, but the lack of a code-state formalism has hindered the study of quantum algorithms in these systems. To address this situation, a code-state formalism for the solution of oracle decision problems in continuously-parameterized quantum systems is developed. Quantum information processing is appealing due its ability to solve certain problems quantitatively faster than classical information processing. Most quantum algorithms have been studied in discretely parameterized systems, but many quantum systems are continuously parameterized. The field of quantum optics in particular has sophisticated techniques for manipulating continuously parameterized quantum states of light, but the lack of a code-state formalism has hindered the study of quantum algorithms in these systems. To address this situation, a code-state formalism for the solution of oracle decision problems in continuously-parameterized quantum systems is developed. In the infinite-dimensional case, we study continuous-variable quantum algorithms for the solution of the Deutsch–Jozsa oracle decision problem implemented within a single harmonic-oscillator. Orthogonal states are used as the computational bases, and we show that, contrary to a previous claim in the literature, this implementation of quantum information processing has limitations due to a position-momentum trade-off of the Fourier transform. We further demonstrate that orthogonal encoding bases are not unique, and using the coherent states of the harmonic oscillator as the computational bases, our formalism enables quantifying the relative performances of different choices of the encoding bases. We extend our formalism to include quantum algorithms in the continuously parameterized yet finite-dimensional Hilbert space of a coherent spin system. We show that the highest-squeezed spin state possible can be approximated by a superposition of two states thus transcending the usual model of using a single basis state as algorithm input. As a particular example, we show that the close Hadamard oracle-decision problem, which is related to the Hadamard codewords of digital communications theory, can be solved quantitatively more efficiently using this computational model than by any known classical algorithm.
Qubits are not the only information carriers that can be used for quantum information processing. In this chapter, we will focus on quantum communication with ‘continuous quantum variables’, or continuous variables for short. In the context of quantum information processing we will also call continuous variables ‘qunats’. We have seen in Chapter 2 that a natural representation of a continuous variable is given by the position of a particle. The conjugate continuous variable is then the momentum of the particle. Unfortunately, the eigenstates of the position and momentum operators are not physical, and we have to construct suitable approximations to these states that can be created in the laboratory. Any practical information processing device must then take into account the deviation of the actual states from the ideal position and momentum eigenstates. Rather than the position and momentum of a particle, we will consider here the two position and momentum quadratures of an electromagnetic field mode. These operators obey the same commutation relations as the canonical position and momentum operators, but they are not the physical position and momentum of field excitations. Approximate eigenstates of the quadrature operators can be constructed in the form of squeezed coherent states. We define a quantum mechanical phase space for quadrature operators, similar to a classical phase space for position and momentum. Probability distributions in the classical phase space then become quasi-probability distributions over the quadrature phase space.We will develop one of these distributions, namely the Wigner function, and identify certain phase-space transformations of the Wigner function with linearoptical and squeezing operations.
The quantum Fourier transform for discrete variable (dvQFT) is an efficient algorithm for several applications. It is usually considered for the processing of quantum bits (qubits) and its efficient implementation is obtained with two elementary components: the Hadamard gate and the controlled--phase gate. In this paper, the quantum Fourier transform operating with continuous variables (cvQFT) is considered. Thus, the environment becomes the Hilbert space, where the natural definition of the cvQFT will be related to rotation operators, which in the NNN--mode are completely specified by unitary matrices of order NNN. Then the cvQFT is defined as the rotation operator whose rotation matrix is given by the discrete Fourier transform (DFT) matrix. For the implementation of rotation operators with primitive components (single--mode rotations and beam splitters), we follow the well known Murnaghan procedure, with appropriate modifications. Moreover, algorithms related to the fast Fourier transform (FFT) are applied to reduce drastically the implementation complexity. The final part is concerned with the application of the cvQFT to general Gaussian states. In particular, we show that cvQFT has the simple effect of transforming the displacement vector by a one-dimensional DFT, the squeeze matrix by a two-dimensional DFT, and the rotation matrix by a Fourier-like similarity transform.
We present a second quantization description of frequency-based continuous variables quantum computation in the subspace of single photons. For this, we define frequency and time operators using the free-field Hamiltonian and its Fourier transform, and we show that these observables, when restricted to the one photon per mode subspace, reproduce the canonical position-momentum commutation relations. As a consequence, frequency and time operators can be used to define a universal set of gates in this particular subspace. We discuss the physical implementation of these gates as well as their effect on single photon states, and show that frequency and time variables can also be used to implement continuous variables quantum information protocols, in the same way than polarization is currently used as a two-dimensional quantum variable.
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.
We obtain sufficient conditions for the efficient simulation of a continuous variable quantum algorithm or process on a classical computer. The resulting theorem is an extension of the Gottesman-Knill theorem to continuous variable quantum information. For a collection of harmonic oscillators, any quantum process that begins with unentangled Gaussian states, performs only transformations generated by Hamiltonians that are quadratic in the canonical operators, and involves only measurements of canonical operators (including finite losses) and suitable operations conditioned on these measurements can be simulated efficiently on a classical computer.