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Demonstration of unconditional one-way quantum computations for continuous variables
Ryuji Ukai, Noriaki Iwata, Yuji Shimokawa, Seiji C. Armstrong, Alberto Politi, Jun-ichi Yoshikawa, Peter van Loock, Akira Furusawa
TL;DR
Photon-based one-way computation demonstrations use probabilistic cluster generation and prepare usual inputs as part of the cluster, limiting deterministic online gates on independently prepared unknown inputs. This work uses a four-mode continuous-variable cluster and homodyne measurements to process independent inputs, demonstrating controllable Fourier and squeezing operations. The scheme applies arbitrary linear transformations to arbitrary one-mode input states in principle.
Problem
Single-photon cluster demonstrations are probabilistic and prepare usual input states within the cluster, limiting deterministic online gates on independently prepared unknown inputs.
Source: p. 1Method
A four-mode continuous-variable linear cluster processes independently prepared inputs through homodyne measurement-based one-way computation.
Source: pp. 1–2Results
The homodyne measurement bases control the demonstrated Fourier and squeezing operations, including x-squeezing levels of 3 dB, 6 dB, and 10 dB.
Source: pp. 3–4Takeaways & Limitations
The scheme can apply arbitrary linear transformations to arbitrary one-mode quantum input states independently prepared from the cluster.
Source: pp. 1, 4Multi-mode transformations are possible by similar means but were not demonstrated in this experiment.
Source: p. 4
Abstract
from arXiv · showhide
Quantum computing promises to exploit the laws of quantum mechanics for processing information in ways fundamentally different from today's classical computers, leading to unprecedented efficiency. One-way quantum computation, sometimes referred to as the cluster model of quantum computation, is a very promising approach to fulfil the capabilities of quantum information processing. The cluster model is realizable through measurements on a highly entangled cluster state with no need for controlled unitary evolutions. Here we demonstrate unconditional one-way quantum computation experiments for continuous variables using a linear cluster state of four entangled optical modes. We implement an important set of quantum operations, linear transformations, in the optical phase space through one-way computation. Though not sufficient, these are necessary for universal quantum computation over continuous variables, and in our scheme, in principle, any such linear transformation can be unconditionally and deterministically applied to arbitrary single-mode quantum states.
Appendix A: Discrete-Variable Cluster Computations
The discrete-variable cluster computation uses a four-qubit linear cluster to apply a sequence of operations to an input qubit. Its computational-basis states are related to the |+⟩ state through the Hadamard gate.
Appendix A: Discrete-Variable Cluster Computations
- A discrete-variable four-qubit linear cluster state is used to demonstrate arbitrary rotations on a qubit [9] [10].The computation applies a sequence of four operations to an input state.Source: Appendix A · p. 4
- The usual input states |+⟩ are prepared as part of the initial cluster state in the cited single-photon experiments.The basis states |0⟩ and |1⟩ are related to |+⟩ through the Hadamard gate.Source: Appendix A · p. 4
- The operations include rotations about the qubit Bloch sphere’s X and Z axes, defined using the Pauli operators X and Z.The rotation operators are denoted Xφi and Zφi.Source: Appendix A · p. 4
Appendix B: Continuous-Variable Cluster Computations
This appendix describes continuous-variable computation using a four-mode linear cluster state as a resource. Homodyne measurement-based operations are combined with teleportation-based coupling to process an input.
Appendix B: Continuous-Variable Cluster Computations
- The resource is a continuous-variable four-mode linear cluster state.Its displayed form uses position and momentum eigenstates across the four modes.Source: Appendix B · p. 4
- The position and momentum eigenstates have real eigenvalues, and the position states form the computational basis for the continuous-variable system.The momentum eigenstates are obtained from position eigenstates by the Fourier transformation.Source: Appendix B · p. 4
- The teleportation-based operation Mtele cannot be decomposed into separate input and mode-1 operations because it uses nonlocal measurements.Both Mtele and M(θi) are elements of the one-mode LUBO transformations.Source: Appendix B · p. 4
Appendix C: Quantum Computation using a Four-mode Linear Cluster State
The appendix derives how input coupling, homodyne measurements, and feed-forward produce a one-mode operation using a four-mode cluster. Finite squeezing leaves errors, while feed-forward removes antisqueezing components from the output noise.
Appendix C: Quantum Computation using a Four-mode Linear Cluster State
- The unknown input is coupled to the four-mode cluster resource using a half beam splitter in a quantum-teleportation process.The input mode is described by its position and momentum quadratures.Source: Appendix C · p. 5
- Homodyne detections measure modes in and 1–3 simultaneously, with all feed-forward postponed until the cluster computation ends.The measurement quadrature angles are denoted θj.Source: Appendix C · p. 5
- In the output-quadrature relation, the first term is the angle-controlled operation, while later terms represent measurement back-actions that feed-forward should eliminate.The remaining error terms arise from the imperfect cluster correlations.Source: Appendix C · p. 5
- Finite squeezing makes the cluster correlations imperfect, introducing additional error components into cluster computations.The scheme assumes equal squeezing levels for the resources; an approximate cluster can be generated from four squeezed vacuum states using a beam splitter and two half beam splitters.Source: Appendix C · pp. 4–5
- Feed-forward eliminates the resource-derived antisqueezing components from the output, considerably reducing extra noise terms.The remaining output noise components derive from the squeezing resources.Source: Appendix C · p. 5
Appendix D: Derivation of Measurement Angles for the Fourier Transformation
The appendix derives measurement settings for implementing the Fourier transformation. It identifies an optimal angle set and relates the input and output while accounting for additional variances.
Appendix D: Derivation of Measurement Angles for the Fourier Transformation
- The Fourier-transformation condition is used to derive the required measurement settings.The derivation treats k1 as a free parameter that can be selected to minimize output error.Source: Appendix D · p. 5
- The free parameter k1 can be chosen to minimize output error, and the optimal parameter set is (kin, k1, k2, k3) = (0, ∞, 0, 0).This set corresponds to measurement angles (θin, θ1, θ2, θ3) = (90°, 0°, 90°, 90°).Source: Appendix D · p. 5
- With the derived input-output relation, the Fourier transformation is achieved and the additional variances are specified.The selected measurement settings are the optimal set stated for the transformation.Source: Appendix D · pp. 5–6
Appendix E: Derivation of Measurement Angles for the x-squeezing Operations
The x-squeezing measurement angles are selected by minimizing the output x-quadrature’s additional variance. The optimum input parameter depends on the squeezing-operation level, and the experiment uses specified angles for each level.
Appendix E: Derivation of Measurement Angles for the x-squeezing Operations
- The optimum kin is independent of resource squeezing levels r and is determined only by the operation level a.It minimizes the additional variance in the output x quadrature.Source: Appendix E · p. 6
- The two choices kin+ and kin− give the same variance, and the experiment selects kin−.Source: Appendix E · p. 6
- For 3, 6, and 10 dB x-squeezing, the selected angle sets are (−42.5°, 62.4°, 63.5°, 76.0°), (−41.4°, 72.2°, 41.9°, 74.4°), and (−47.7°, 79.2°, 25.9°, 78.4°), respectively.The angle tuples are ordered (θin, θ1, θ2, θ3).Source: Appendix E · p. 6
Appendix F: Remarks on Fidelity and Scalability
The appendix relates finite-squeezing noise to fidelity and scalability in measurement-based continuous-variable computation. It gives resource-squeezing and computation-length estimates while noting that strict fault-tolerant scalability remains unproven.
Appendix F: Remarks on Fidelity and Scalability
- Operation fidelity is defined as overlap between the ideal pure output and the experimental output density matrix.For a Fourier transformation on an arbitrary pure Gaussian input, the discussion derives fidelity from output variances and excess noise after four elementary measurement-based steps.Source: Appendix F · p. 6
- An n-step cluster teleportation roughly corresponds to an n/4-step sequential teleportation, and the appendix extends its fidelity discussion to n-step computations.Source: Appendix F · p. 6
- ne^-2r is the approximate excess noise for an n-step one-way computation, so maintaining output fidelity requires resource squeezing e+2r to grow roughly in proportion to n.The passage also states that any desired accuracy can be achieved for an arbitrarily large cluster with the same squeezing levels when connectivity is constant.Source: Appendix F · pp. 6–7
- Cluster accuracy can remain independent of cluster size at fixed squeezing and constant connectivity, while computation accuracy decreases with computation length.Connectivity is the maximum number of nearest neighbors for any mode.Source: Appendix F · p. 7
- At about −5.5 dB squeezing, the estimate is 13 teleportation steps; at −12.7 dB, it is 73 steps, using F0 = 1/2 as the classical-limit benchmark.The authors characterize this as scalability in a weak sense.Source: Appendix F · p. 7
- Strict scalability for continuous-variable computation lacks a threshold-theorem result, and discrete-variable error models do not directly apply to realistic probabilistic PDC cluster preparation.Source: Appendix F · p. 7
- Finite squeezing intrinsically adds measurement-based computation errors, and no known error-correction scheme suppresses them for a logical continuous-variable state.The authors state that such an encoded scheme must use a nonlinear, non-Gaussian element.Source: Appendix F · pp. 7–8
- Maintaining output entanglement above a fixed bound requires initial squeezing to increase logarithmically with the maximum number of operations.The corresponding initial resource entanglement has the same logarithmic scaling.Source: Appendix F · pp. 7–8