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arXiv:quant-ph/0410204 · Physical Review A

Coherent-state linear optical quantum computing gates using simplified diagonal superposition resource states

Austin P. Lund, Timothy C. Ralph

Published Mar 8, 2005• 9 citations
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Abstract

In this paper we explore the possibility of fundamental tests for coherent-state optical quantum computing gates [T. C. Ralph et al., Phys. Rev. A 68, 042319 (2003)] using sophisticated but not unrealistic quantum states. The major resource required in these gates is a state diagonal to the basis states. We use the recent observation that a squeezed single-photon state $[S(r)\ensuremath{\mid}1⟩]$ approximates well an odd superposition of coherent states $(\ensuremath{\mid}\ensuremath{\alpha}⟩\ensuremath{-}\ensuremath{\mid}\ensuremath{-}\ensuremath{\alpha}⟩)$ to address the diagonal resource problem. The approximation only holds for relatively small $\ensuremath{\alpha}$, and hence these gates cannot be used in a scalable scheme. We explore the effects on fidelities and probabilities in teleportation and a rotated Hadamard gate.

DOI
10.1103/physreva.71.032305

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In this paper we explore the possibility of fundamental tests for coherent-state optical quantum computing gates [T. C. Ralph et al., Phys. Rev. A 68, 042319 (2003)] using sophisticated but not unrealistic quantum states. The major resource required in these gates is a state diagonal to the basis states. We use the recent observation that a squeezed single-photon state [S(r)\ensuremath∣1⟩][S(r)\ensuremath{\mid}1⟩][S(r)\ensuremath∣1⟩] approximates well an odd superposition of coherent states (\ensuremath∣\ensuremathα⟩\ensuremath−\ensuremath∣\ensuremath−\ensuremathα⟩)(\ensuremath{\mid}\ensuremath{\alpha}⟩\ensuremath{-}\ensuremath{\mid}\ensuremath{-}\ensuremath{\alpha}⟩)(\ensuremath∣\ensuremathα⟩\ensuremath−\ensuremath∣\ensuremath−\ensuremathα⟩) to address the diagonal resource problem. The approximation only holds for relatively small \ensuremathα\ensuremath{\alpha}\ensuremathα, and hence these gates cannot be used in a scalable scheme. We explore the effects on fidelities and probabilities in teleportation and a rotated Hadamard gate.
Reliable entangling gates for qubits encoded in single-photon states represent a major challenge on the road to scalable quantum computing architectures based on linear optics. In this work, we present two approaches to develop high-fidelity, near-deterministic controlled-sign-shift gates based on the techniques of quantum gate teleportation. On the one hand, teleportation in a discrete-variable setting, i.e., for qubits, offers unit-fidelity operations but suffers from low success probabilities. Here, we apply recent results on advanced linear optical Bell measurements to reach a near-deterministic regime. On the other hand, in the setting of continuous variables, associated with coherent states, squeezing, and, typically, Gaussian states, teleportation can be performed in a deterministic fashion, but the finite amount of squeezing implies an inevitable deformation of a teleported single-mode state. Using a new generalized form of the nonlinear-sign-shift gate for gate teleportation, we are able to achieve fidelities of the resulting controlled-sign gate above 9090%90. A special focus is also put on a comparison of the two approaches, not only with respect to fidelity and success probability, but also in terms of resource consumption.
Quantum computing using two coherent states as a qubit basis is a proposed alternative architecture with lower overheads but has been questioned as a practical way of performing quantum computing due to the fragility of diagonal states with large coherent amplitudes. We show that using error correction only small amplitudes (alpha>1.2) are required for fault-tolerant quantum computing. We study fault tolerance under the effects of small amplitudes and loss using a Monte Carlo simulation. The first encoding level resources are orders of magnitude lower than the best single photon scheme.
We present an optical scheme to conditionally generate even or odd squeezed superpositions of coherent states (SSCSs). The optical setup consists of an unbalanced beam splitter whose transmittance tends to unity, and additional balanced beam splitters and photodetectors in auxiliary modes. Squeezed coherent states with different amplitudes are the input states in the optical scheme. The single-qubit operations are probabilistic and employ two- and three-photon subtractions from initial beams as the driving force. Generation of the even or odd SSCSs is observed in a wide diapason of values of used parameters. We consider a possibility to realize a one-way Hadamard gate for the squeezed coherent states when the base states are transformed into superposition states. States approximating the output states of a Hadamard gate with high fidelity can be realized by imposing restrictions on the values of used parameters. Higher-order subtractions from input beams are necessary to generate the SSCSs with larger amplitudes and higher fidelities. The problem is resolved in a Wigner representation to take into account imperfections of the optical devices.
In recent years coherent state qubits, |α〉 ± | - α〉 have gained much interest. The main reason is the promise to realize a universal set of coherent state quantum gates with a relative low overhead using, only linear optics and photon counters. A universal set of gates consists of a single mode and a two mode phase gate and a Hadamard gate. The Hadamard is the gate that transforms one set of basis states into the conjugate basis, e.g | ± α〉 - | - α〉. Here we present an experimental demonstration of a Hadamard gate for coherent state qubits. The implementation is based on the scheme proposed by Marek and Fiurasek (2010).
We suggest an efficient scheme for quantum computation with linear optical elements, where the qubits are encoded in single photon states. The scheme reduces the resources required per logical gate by several orders of magnitude, compared to an earlier proposal of Knill, Laflamme, and Milburn, while the resource overhead per gate is independent of the length of the computation. A central feature of the scheme, enabling these improvements, is the prior construction of a "linked" photon state designed according to the particular quantum circuit one wishes to process. Once this state has been successfully prepared, the computation is pursued deterministically by a sequence of teleportation steps.
Quantum information and quantum computing offer a new range of possibilities in terms ofnew computing algorithms, the ability to perform new types of operations and make statementsabout information in terms of what it is physically possible to know about a system.Experiments in quantum information and quantum computing have been advancing significantly,especially over the last decade. In optical quantum computing there are two maintypes of experiments that are performed today. They are experiments involving single photonswith information encoded on the polarisation of the photon and experiments involvingstates with Gaussian noise profiles in their quadratures. There appears to be no way toperform the required operations for universal quantum computing using operations whichtake Gaussian states to Gaussian states. Universal quantum computing using single photonsis possible but uses either enormously large non-linearities or a large number of resources.Also, operations which take Gaussian states to non-Gaussian states are not well understoodin general and can often be very difficult to implement.This thesis presents three sources of optical non-Gaussian states which demonstrate someof the interesting properties of these states. First, a scheme for universal quantum computingusing superpositions of coherent states is presented. Error correction and the fault toleranceproperties of this scheme is also explored and compared with current single photon architectures.These gates require superpositions of coherent states as resources and a relativelysimple scheme is shown for how these states may be generated. Next a scheme for distillingnear ideal Gaussian entanglement using non-Gaussian operations is presented. The loss andnoise conditions under which ideal distillation can be achieved are presented. Finally, a verysimple way of generating mixed non-Gaussian states from two Gaussian states is presented.These states are shown to give rise to counter-intuitive results when measuring their entanglementand when analysing their operational ability in terms of teleportation and quantumkey distribution.