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Frequency-encoded photonic qubits for scalable quantum information processing

Joseph M. Lukens, Pavel Lougovski

arXiv:1612.03131v1quant-ph

TL;DR

The paper addresses the difficulty of interconnecting heterogeneous qubits when conventional approaches require frequency-matched photons. It develops spectral LOQC, which uses dual-rail frequency encoding and telecommunications-compatible pulse shapers and modulators. The resulting platform theoretically supports universal computation with linear optical-resource scaling and reconfigurable, fiber-compatible operation.

  • Problem

    Conventional photonic interconnects and spatial LOQC are poorly suited to heterogeneous qubits because they require frequency compatibility and are incompatible with robust single-mode fiber.

  • Method

    The paper develops spectral LOQC using dual-rail frequency encoding, Fourier-transform pulse shapers, electro-optic modulators, and alternating decompositions of spectral-mode transformations.

  • Results

    Universal spectral LOQC uses O(N) optical components for arbitrary unitary transformations on N spectral modes, while the two-qubit spectral CZ gate reaches success probabilities within 1% of the KLM limit at R = 4.

  • Takeaways & Limitations

    Frequency-encoded photons can serve as reconfigurable quantum interconnects compatible with long-distance fiber networks and telecommunications technology.

Abstract

from arXiv · show

Among the objectives toward large-scale quantum computation is the quantum interconnect: a device which uses photons to interface qubits that otherwise could not interact. However, current approaches require photons indistinguishable in frequency---a major challenge for systems experiencing different local environments or of different physical compositions altogether. Here we develop an entirely new platform which actually exploits such frequency mismatch for processing quantum information. Labeled "spectral linear optical quantum computation" (spectral LOQC), our protocol offers favorable linear scaling of optical resources and enjoys an unprecedented degree of parallelism, as an arbitrary $N$-qubit quantum gate may be performed in parallel on multiple $N$-qubit sets in the same linear optical device. Not only does spectral LOQC offer new potential for optical interconnects; it also brings the ubiquitous technology of high-speed fiber optics to bear on photonic quantum information, making wavelength-configurable and robust optical quantum systems within reach.

I. INTRODUCTION

Photons are promising quantum interconnects because they can transmit quantum information between separated qubits, but conventional LOQC struggles with heterogeneous systems and fiber transmission. Spectral LOQC addresses these constraints by encoding qubits in frequency rather than spatial modes.

  • Photons can interface separated quantum systems because they operate across temperatures, experience little decoherence, are optically manipulated, and travel at light speed.
  • Quantum interconnects require optical processing between heterogeneous qubits that may be thermally, spectrally, or spatially incompatible.
  • Conventional spatial LOQC is poorly suited to heterogeneous interconnects because it is unavailable in single-mode fiber and uses a single spectral mode.
  • Spectral LOQC encodes photonic qubits in two discrete spectral modes and operates them with Fourier-transform pulse shapers and electro-optic phase modulators.

II. PROTOCOL COMPONENTS

The protocol uses monochromatic frequency bins as dual-rail qubits and manipulates them with telecommunications-compatible components. This frequency-domain approach offers practical measurement and synchronization advantages over ultrafast temporal modes.

  • Earlier time-frequency proposals relied on ultrafast modes requiring sophisticated nonlinear or polarization-based control, whereas spectral LOQC uses narrowband modes.
  • Monochromatic frequency bins provide spectral modes analogous to those used in dense wavelength-division multiplexing.
  • Spectral modes support high-resolution measurement, reduce detector-jitter constraints, and permit temporally simultaneous processing of multiple modes.
  • A dual-rail qubit occupies two spectral modes separated by a fixed frequency spacing, with the logical state determined by the photon’s mode.
  • Fourier-transform pulse shapers separate frequency modes and independently manipulate their phases using programmable components.
  • Electro-optic phase modulators generate frequency-comb sidebands through electrically controlled temporal phase modulation.

PULSE SHAPER

Spectral LOQC combines diagonal phase operations from pulse shapers with global frequency-mode mixing from electro-optic modulators. Alternating these components can realize arbitrary unitary transformations with linear optical-resource scaling.

  • PULSE SHAPER: A Fourier-transform pulse shaper applies programmable phases independently to separated spectral modes.
  • PULSE SHAPER: Electro-optic modulation mixes frequency rails globally, allowing amplitudes in modes A0 and A1 to interfere like a spatial interferometer.
  • PULSE SHAPER: Alternating pulse shapers and EOMs reproduces arbitrary unitary transformations on N spectral modes using O(N) components.
  • PULSE SHAPER: Spectral LOQC’s linear component scaling contrasts with O(N^2) scaling for spatial- or polarization-encoded LOQC.
  • PULSE SHAPER: EOMs provide a simpler scalable alternative to nonlinear-medium spectral mixing because they use one electrical control and produce no noise photons from powerful optical fields.

III. DERIVING UNVERISAL GATE SET

Spectral LOQC realizes a universal gate set using frequency-mode transformations from pulse shapers and electro-optic modulators. The Hadamard gate is deterministic, while the heralded cz gate approaches the best known two-ancilla linear-optical success probability.

  • Gate construction: Spectral LOQC assumes ancillary single photons, photon-number-resolving detectors, and vacuum modes for nondestructive two-qubit-gate heralding.The optical network alternates pulse shapers and EOMs, with each element modeled over a truncated spectral mode space.
  • Gate construction: The network transformation V is built from R alternating pulse-shaper and EOM pairs acting on computational and ancilla spectral modes.Gate performance is evaluated by deriving the projected Fock-basis state transformation and comparing it with the target using fidelity and success probability.
  • Single-qubit gates: The Hadamard gate achieves F = 1 and P = 1 with two pulse shapers and two EOMs, requiring no ancilla photons or heralding detectors.A single EOM cannot realize the required lossless mixer, but two EOMs suffice.
  • Two-qubit gate: The cz gate uses two ancilla photons in auxiliary spectral modes, followed by R pulse-shaper/modulator pairs and spectrally resolved detection of a heralding coincidence pattern.The controlled-Z operation completes the universal gate set together with phase and Hadamard operations.
  • Two-qubit gate: At R = 4, the cz success probability comes within 1% of the KLM limit P = 2/27 ≈0.0741, while R = 3 exceeds the original KLM value P = 1/16 = 0.0625.The optimization constrains fidelity to F ≥0.9999 while maximizing success probability.

IV. PRACTICAL CONSIDERATIONS

Practical spectral LOQC requires sufficient bandwidth, spectral resolution, modulation speed, and low-loss components. Simulations and existing telecommunications hardware indicate favorable bandwidth scaling and substantial parallelism, while insertion loss remains a central implementation challenge.

  • Bandwidth scaling: The Hadamard gate reaches 99% fidelity with a six-mode passband and 90% of asymptotic success probability with an eight-mode band.The required optical passband is therefore about eight times the logical mode spacing for good performance.
  • Bandwidth scaling: For cz gates, increasing R raises asymptotic success probability and improves effective-mode efficiency; R = 4 reaches the 0.0741 linear-optical benchmark using an 18-mode effective bandwidth.Figure 3 compares fidelity and success probability versus optical bandwidth for R = 2, 3, and 4.
  • Implementation requirements: The linear network must provide spectral resolution, microwave bandwidth, and low loss, whose engineering requirements can conflict when choosing frequency spacing and component resolution.Wider mode spacing eases spectral resolution but increases modulation-bandwidth demands, while high-resolution shapers can introduce substantial loss.
  • Parallel operation: With 10-GHz mode spacing and 500 available channels, an 18-mode R = 4 cz gate could support approximately 28 gates in parallel without increasing device count.The required electronics are about 50-GHz bandwidth, and commercial modulators exceeding 100 GHz are available.
  • Experimental architecture: A proposed experiment separates ancilla preparation, the pulse-shaper/EOM linear network, and ancilla detection for heralding successful operation.Ancillas can be generated by spontaneous four-wave mixing in a resonator and selected through frequency-resolved detection.
  • Loss and integration: Discrete components incur a few decibels of insertion loss per element, whereas demonstrated AWG-based shapers range from commonly greater than 10 dB to approximately 0.5 dB in favorable designs.On-chip implementations may reduce coupling loss and improve scalability, but loss remains a practical boundary.

V. CONCLUSIONS AND FUTURE DIRECTIONS

The paper proposes a universal linear-optical quantum-computing platform using dual-rail frequency encoding and telecommunications-compatible components. It highlights reconfigurability, circuit-synthesis opportunities, and direct interconnection of frequency-disparate systems.

  • Spectral LOQC is a universal platform based on dual-rail frequency encoding that requires no optical nonlinearities or spatial interferometers.
  • Programmable pulse shapers and EOMs can reconfigure one physical arrangement into several quantum circuits by changing electrical controls.
  • Fourier-dual phase manipulation suggests computation-specific optical circuit synthesis rather than assembling only one- and two-qubit gates.
  • Spectral qubits directly interface with frequency-disparate systems, support long-distance optical-fiber transmission, and use telecommunications technology.

Appendix A: Optimization Procedure

The optimization procedure converts programmable spectral and temporal phase operations into photonic state transformations, then compares those transformations with target gates using fidelity and success probability.

  • The mode transformation is modeled as alternating spectral and temporal diagonal phase operations separated by DFT matrices.The D_k and ˜D_k matrices represent spectral and temporal phase modulation, respectively.
  • The procedure converts the mode transformation into a state-space matrix over input and output photonic basis states.For a single qubit, both input and output spaces have dimensionality 2.
  • Two-qubit transformations account for multiple photon pathways, with input dimension 4 and output dimension 10 when computational photons may occupy the same mode.The two-qubit example includes two ancillary photons in modes u and v.
  • The resulting state transformation W is compared with the target matrix T using Hilbert-Schmidt fidelity.The target matrix is obtained by mapping basis labels to logical basis states.
  • Optimization varies the 2RM phases so that fidelity reaches 1 while success probability is maximized.

Appendix B: Full Solutions

The appendix records numerical searches and phase solutions for the optimized Hadamard and controlled-Z gates. It also explains how the plotted EOM and pulse-shaper phases relate to implementation requirements.

  • Numerical searches using MATLAB’s Optimization Toolbox produced the results summarized in Table 1.The searches are described as thorough but nonexhaustive.
  • The appendix records specific pulse-shaper and EOM phases for optimal Hadamard and controlled-Z gates.
  • The simulations truncate the spectral-mode space at M = 128 modes, described as sufficiently precise without making computation intractable.
  • For the Hadamard solution, microwave power spectra indicate the electronic bandwidth required by an arbitrary waveform generator, while pulse-shaper phases are individually mode-resolved.

EOM 2 EOM 3 EOM 4

Figure 7 presents the phases used to realize a spectral controlled-Z gate with R = 4, using the same phase and spectrum layout described for the preceding controlled-Z solutions.

  • The gate is a spectral controlled-Z operation with R = 4.
  • The figure presents temporal phases over one period and corresponding microwave power spectra for two EOMs.
  • The figure also presents pulse-shaper phases for the gate implementation.
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