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Photonic architecture for scalable quantum information processing in NV-diamond

Kae Nemoto, Michael Trupke, Simon J. Devitt, Ashley M. Stephens, Kathrin Buczak, Tobias Nobauer, Mark S. Everitt, Jorg Schmiedmayer, William J. Munro

arXiv:1309.4277v1quant-ph

TL;DR

Quantum processors require architectures that combine error correction with feasible hardware despite unavoidable decoherence and other errors. This paper develops fiber-connected optical-cavity modules containing single NV− centres, using electron spins, nuclear-spin memories, and photons to generate topological cluster states. The architecture tolerates approximately 0.73% error per elementary gate at the circuit level and is reported to be consistent with present technology, while current implementations remain probabilistic but heralded.

  • Problem

    Quantum systems inevitably suffer decoherence and other errors, so a reliable and scalable quantum computer requires architecture integrating quantum error correction with feasible experimental technology.

  • Method

    The paper develops fiber-connected optical-cavity modules containing single NV− centres, with electron spins interfaced to photons and long-lived nuclear-spin memories for cluster-state processing.

  • Results

    0.73% is the maximum tolerable error per elementary quantum gate at the circuit level, and physical-level analysis finds the architecture consistent with present technology.

  • Takeaways & Limitations

    The modules and their two-dimensional connectivity support topological cluster-state error correction for large-scale quantum information processing.

  • Takeaways & Limitations

    Current limitations make processes probabilistic but heralded, with photon loss reducing successful detection probability and repeated attempts needed to establish links.

Abstract

from arXiv · show

Physics and information are intimately connected, and the ultimate information processing devices will be those that harness the principles of quantum mechanics. Many physical systems have been identified as candidates for quantum information processing, but none of them are immune from errors. The challenge remains to find a path from the experiments of today to a reliable and scalable quantum computer. Here, we develop an architecture based on a simple module comprising an optical cavity containing a single negatively-charged nitrogen vacancy centre in diamond. Modules are connected by photons propagating in a fiber-optical network and collectively used to generate a topological cluster state, a robust substrate for quantum information processing. In principle, all processes in the architecture can be deterministic, but current limitations lead to processes that are probabilistic but heralded. We find that the architecture enables large-scale quantum information processing with existing technology.

I. INTRODUCTION

The paper proposes a scalable quantum-computing architecture that integrates quantum error correction with feasible NV−-diamond technology. Its fiber-connected modules support topological cluster-state correction, with a circuit-level tolerable gate error of approximately 0.73%.

  • Quantum systems inevitably suffer decoherence and other errors, motivating architectures that combine error correction with feasible experimental technology.
  • An architecture based on optical-cavity modules containing single NV− centres connects modules through photons in a fiber-optical network.
  • A regular two-dimensional array of modules, detectors, and classical control lines provides connectivity for topological cluster-state error correction independently of network size.
  • 0.73% is the maximum tolerable error per elementary quantum gate at the circuit level.
  • Physical-level analysis indicates that the required module-component performance is consistent with present technology and might be achievable in the near future.

II. FUNDAMENTAL BUILDING BLOCKS

The fundamental building blocks use state-dependent photon reflection to distribute entanglement between emitter-cavity modules. Although the protocol is probabilistic, repeat-until-success operation provides high-probability links while preserving extremely high fidelity.

  • The fundamental module is a general emitter-cavity system whose state-dependent reflectivity underlies the entanglement-distribution scheme.
  • A Michelson-interferometer arrangement connects two modules with an optical fiber, and dark-port detection projects the emitters into a singlet Bell state.
  • p = η2/8 is the entanglement-generation success probability, where η2 includes source, detector, and transmission losses.
  • The generated entangled state has fidelity above 99%, and cavity-reflection imbalance slightly reduces success probability without degrading fidelity.
  • Repeat-until-success operation establishes an entanglement link with high probability despite the low success probability of each attempt.
  • A memory qubit coupled to the cavity system allows the electron-spin interface to be reused for entanglement creation with additional modules and cluster-state generation.

III. THE DIAMOND MODULE

The concrete module implementation uses an NV− electron spin for optical interfacing and a 15N nuclear spin as a memory. Magnetic-field tuning and spin-control sequences support selective qubit operation and repeated entanglement attempts.

  • A fiber-connected optical cavity containing one NV− centre supplies the concrete module implementation.
  • A magnetic field of approximately 20 mT separates the electron-spin levels so |0⟩ and |+1⟩ form the qubit, while 15N provides a spin-1/2 nuclear memory.
  • The electron spin is prepared in |0⟩ before the protocol, and rotating it into a superposition activates the hyperfine-coupling clock.
  • Spin-echo sequences decouple electron and nuclear spins when entanglement distribution fails before electron-spin reinitialization and another attempt.
  • The NV− optical states Ex and Ey reduce sensitivity to rogue strain or electric-field influences in the y-direction, while Ex can tune sites into resonance.

A. Quantum non-demolition detection

The module performs quantum non-demolition detection of the NV− state through conditional photon reflection, a qubit flip, and a second photon measurement.

  • The non-demolition measurement sequence consists of a photon measurement, a microwave π-pulse qubit flip, and a second photon measurement.
  • A photon is reflected and detected when the NV− centre is in |0⟩ and is otherwise lost, enabling state-sensitive readout.

B. Remote entanglement

Remote entanglement is generated by heralded photon detection between electron spins and then transferred to long-lived nuclear-spin memories through hyperfine coupling. Timing the interaction at specific points in its oscillation enables controlled entanglement transfer and protects the memories during retries.

  • B. Remote entanglement: Electron spins are initialized in equal superpositions, and photon detection at the interferometer’s dark port heralds successful entanglement.The procedure repeats until success, enabled by the cycling properties of the NV− transition.
  • B. Remote entanglement: The nuclear spins serve as long-lived memories that store cluster-state nodes after entanglement is transferred from the electrons.The transfer uses the Ising component of the hyperfine interaction.
  • B. Remote entanglement: At the π point of the hyperfine oscillation, the electron–nuclear interaction implements a controlled-phase gate, while the 2π point implements identity.The entanglement oscillates from zero to maximum over time.
  • B. Remote entanglement: The complete protocol polarizes both spins, creates electron–nuclear correlations with rotations and hyperfine coupling, and measures the electron to initialize the nuclear spin.The nuclear spin is initialized into the |n+⟩ state before subsequent entanglement operations.

IV. SHARING ENTANGLED STATES BETWEEN THREE MODULES

The architecture extends entanglement from two nuclear-spin memories to larger cluster states by repeatedly creating electron–electron bonds and transferring them to nuclear spins. Accurate timing during failed attempts prevents photon-loss feedback from corrupting previously stored entanglement.

  • IV. SHARING ENTANGLED STATES BETWEEN THREE MODULES: A new electron–electron bond between modules B and C is created with the repeat-until-success protocol and transferred to nuclear spins.The existing entangled pair resides in modules A and B, while only module C’s nuclear spin is initialized during extension.
  • IV. SHARING ENTANGLED STATES BETWEEN THREE MODULES: The extension must preserve the high-fidelity entangled state already stored in modules A and B while module B creates new entanglement with module C.Photon loss could otherwise feed back through the permanent hyperfine coupling and introduce catastrophic errors.
  • IV. SHARING ENTANGLED STATES BETWEEN THREE MODULES: During retries, the protocol waits until the 2π point, decoupling electron and nuclear spins and protecting stored nuclear information from feedback errors.After successful electron entanglement, waiting until the π point transfers the bond to the nuclear spins.
  • IV. SHARING ENTANGLED STATES BETWEEN THREE MODULES: Time-sequenced entangling and spin-echo-like decoupling avoid photon-loss-induced decoherence while the electron spins are reinitialized.The hyperfine sequence clock begins when the photonic entangling protocol rotates the electron out of |0⟩.
  • IV. SHARING ENTANGLED STATES BETWEEN THREE MODULES: Repeating the procedure across modules generates arbitrary cluster states, including a three-dimensional topological cluster state for fault-tolerant computation.Its fundamental dagger-shaped unit contains five modules and requires four entangling steps.

V. BENCHMARKING THE PHOTONIC ARCHITECTURE

The architecture is benchmarked by its fault-tolerance threshold, component error requirements, and computational cycle time. The analysis finds a 0.73% threshold, sub-order-of-magnitude improvement requirements for several parameters, and logical-gate rates constrained chiefly by optical connections and nuclear-spin operations.

  • V. BENCHMARKING THE PHOTONIC ARCHITECTURE: Preparing each topological-cluster layer in five rather than six circuit steps raises the error threshold to 0.73%.The target error rate for the five relevant gates is approximately 0.1%.
  • V. BENCHMARKING THE PHOTONIC ARCHITECTURE: The component analysis treats nuclear and electron decoherence, measurement and rotation efficiency, timing error, and photon absorption as physical error sources.Each plotted curve assumes one non-zero error apart from possible photon absorption errors.
  • V. BENCHMARKING THE PHOTONIC ARCHITECTURE: Required improvements for the remaining parameters are less than one order of magnitude and are not limited by known fundamental limitations of NV− systems.Electron and nuclear decoherence are already sufficiently low, while other parameters still require improvement.
  • V. BENCHMARKING THE PHOTONIC ARCHITECTURE: For P = 99.9% bond connections and pc = 6.25%, a nuclear–nuclear bond takes 3.5 µs and a cluster unit cell is prepared every approximately 30 µs.The five-step circuit constructs each cross-sectional layer of the topological cluster state.
  • V. BENCHMARKING THE PHOTONIC ARCHITECTURE: A logical CNOT with pc = 6.25% takes 3.4 ms, corresponding to a clock frequency of approximately 295 Hz.The rate can improve with better optical efficiencies but is ultimately limited by the NV− hyperfine interaction for nuclear-spin operations.
  • V. BENCHMARKING THE PHOTONIC ARCHITECTURE: With deterministic electron–electron connections, a logical CNOT would take approximately 960 µs, or approximately 1 kHz, and become limited by nuclear measurement.This removes the probabilistic connection bottleneck from the cycle time.

VI. DISCUSSION

The architecture distributes NV-based modules through optical links and uses compact, two-layer cluster-state operation for scalable processing. Its performance depends on coherence, heralded entanglement, and measurement requirements that are broadly consistent with present technology but still need improvement.

  • Cluster-state operation: Only two successive layers of the three-dimensional topological cluster state need to be prepared and stored at any time.Measuring one layer teleports computation to the other, allowing arbitrarily deep computation with a fixed number of physical qubits.
  • Performance and scaling: With P = 99.0%, the number of attempts can be reduced to s = 71 for pc = 6.25%.The architecture uses probabilistic entangling bonds repeated until success, with the required success probability depending on the communication setting.
  • Experimental requirements: The architecture’s physical requirements are broadly consistent with present technology, but measurement efficiency still requires improvement.More sophisticated adaptive error analysis may reduce the stringency of some physical requirements.
  • Coherence properties: The electron-spin T2* is the limiting coherence parameter, although the controlled-phase gate error is small in principle (< 10−5).The stated operating assumptions are isotopically pure diamond and temperatures of 4–20 K.
  • Architecture: Optical-cavity NV− modules connect through photons, while nuclear spins provide long-lived memories for storing and processing quantum information.The architecture also includes photon detectors and classical control lines arranged in a regular two-dimensional array.

2. Quantum non-demolition measurement of the electron spin state

The architecture uses state-dependent cavity reflection to perform near-nondemolition electron-spin measurements, with repeated photon probes mitigating loss. High cooperativity and suitable optical efficiency are required because spin flips and photon loss limit performance.

  • Measurement principle: State-dependent cavity reflection enables quantum non-demolition measurement and electron-spin initialisation using repeated single-photon probes.The sequence infers the spin from photon detection patterns while allowing repeated measurements when photons are lost.
  • Measurement principle: High cooperativity makes the empty cavity weakly reflecting while an NV− centre in |0⟩ approaches unit reflection, improving measurement contrast.In the high-cooperativity limit, emitter excitation decreases as PS →1/C, although off-resonant excitation of |+1⟩ remains.
  • Implementation choices: The low-cooperativity implementation is unsuitable for the main scheme because spin-flip-inducing transitions occur at experimentally observed rates of about 1%.The authors retain it for initial demonstrations and focus on the high-cooperativity implementation.
  • Measurement principle: A detector click strongly indicates the NV− centre is in |0⟩, while the sequence can detect leakage into |−1⟩.The model uses Pflip,0 = 0.003 and Pflip,+1 = 0.35, although the passage notes no consensus on these values.
  • Performance and errors: A single QND measurement has an error rate of about 1%, exceeding the required 7.3×10^-3, motivating multiple measurements in the high-cooperativity approach.The architecture requires optical efficiency η2 ≳30%, and a useful working cooperativity is C ∼50.
  • Performance and errors: Photon loss lowers the probability of detecting a probe and arises from absorption, scattering, coupling inefficiency, and detector inefficiency.The loss parameter η2 ranges from 0 to 1, with η2 = 1 representing no loss.

3. Entanglement

Remote NV− centres are entangled by sending a single photon through a beamsplitter-linked pair of cavities and heralding on dark-port detection. The resulting entanglement can be transferred to nuclear memories, while repeat-until-success operation addresses low success probability.

  • Remote electron entanglement: A single photon sent through a 50:50 beamsplitter interacts with two remote cavity-NV− modules, and dark-port detection projects their electron spins into an entangled state.The modules are arranged in a Michelson interferometer configuration, with the photon returning through the beamsplitter after cavity interactions.
  • Remote electron entanglement: Imperfect reflection and transmission reduce the entangled-state amplitude, whereas differences between the two cavities generally reduce fidelity.A small loss element can compensate unequal reflected amplitudes while preserving the required state form.
  • Heralding and repetition: Photon loss leaves the electron state indeterminate, so the protocol repeats attempts until an entanglement link is established with high probability.Loss can occur in the channel, at the NV− centres, through imperfect coupling, or through inefficient detection.
  • Transfer to nuclear memories: Electron-spin and nuclear-spin operations use microwave rotations and hyperfine interactions, including a natural CPHASE gate with tmax ∼165 ns.The nuclear spin serves as a long-lived memory coupled to the electron spin through the hyperfine interaction.
  • Transfer to nuclear memories: Upon successful electron entanglement, the protocol transfers it to the two nuclear spins and circumvents photon-loss-induced decoherence through the hyperfine interaction.Accumulation errors from many attempts affect the nuclear–nuclear CPHASE gate and must be strongly suppressed.
  • Architecture-level operation: Approximately 10^7 attempts yield a success probability P = 0.999, effectively deterministic for the stated purpose.The architecture combines remote entanglement, spin rotations, and nuclear-spin operations for cluster-state preparation and distributed processing.

Appendix E: Electron–electron connection

The architecture’s remote electron–electron connections accumulate timing, decoherence, and excitation-related errors across probabilistic attempts. The protocol maps these connections into remote nuclear-spin entanglement and can prepare arbitrary cluster states by repetition.

  • Error sources: Three error sources accumulate during remote electron-spin entanglement: hyperfine timing errors, nuclear decoherence, and electronic-system excitation.Timing errors propagate Z errors to nuclei, while excitation attempts can fail and occasionally induce nuclear-spin errors.
  • Connection performance: The connection probability is pc = 0.125po for the single-sided-cavity protocol.Table I relates required timing accuracy and attempt counts to target connection probabilities P = 0.99 and P = 0.999.
  • Error mapping: The general error mapping distinguishes photon loss, relaxation to |0⟩, and relaxation to | + 1⟩, with the latter associated with possible nuclear-spin errors.The worst-case nuclear error depends on when the electron decays relative to the 165 ns hyperfine-coupling π point.
  • Cluster-state generation: Repeating the remote-entanglement protocol implements an effective CPHASE gate between nuclear spins and can generate arbitrary cluster states.The nuclear spins serve as long-lived memories, and repeating the protocol across modules produces cluster states suitable for fault-tolerant computation.

1. Topological cluster-state error correction

Topological cluster-state error correction uses a three-dimensional structure in which two dimensions encode spatial logical qubits and the third represents computation time. The architecture prepares only adjacent layers and uses optically mediated connections to support a two-dimensional module array.

  • Physical unit cell: A physical unit cell contains two qubit layers, whose two-dimensional projection requires nearest- and next-nearest-neighbour interactions.The projected layout preserves the unit cell’s colour coding while adding next-nearest-neighbour connections.
  • Topological cluster structure: Three-dimensional topological cluster states reserve two dimensions for protected logical qubits and use the third as the temporal computation axis.Only two adjacent layers need to be prepared at any given time.
  • Preparation circuit: The two-layer preparation circuit uses six steps because an optimal five-step circuit is impossible with only two layers of modules.Modules remain idle for one step after measurement; the idle error is restricted to nuclear decoherence over the relevant timescale.
  • Synchronization: Simultaneous connections are grouped into synchronized steps, requiring some modules to wait until all connections in the current step succeed.The architecture considers both synchronous and asynchronous operation modes.
  • Connection attempts: For pc = 6.25%, g = 107 attempts are required to establish a bond with probability P = 0.99%.The attempt count follows g = log(1 −P)/ log(1 −pc); the main text also uses the average 1/pc for operation-rate estimates.
  • Missing-bond tolerance: Heralded failed connections permit missing bonds, but increasing their fraction lowers the threshold for other errors.The calculations do not exploit this robustness or quantify the tradeoff between missing bonds and other error rates.

Appendix G: Experimental requirements and expected performance

The architecture is evaluated against topological cluster-state error-correction requirements using operation-error estimates that account for repeated probabilistic connections. The reported parameters place measurement and initialization below target, while CPHASE errors remain above target but below threshold.

  • Threshold: The topological cluster-state threshold is approximately 0.73% for measurement/initialization and CPHASE errors during state preparation.This threshold is used as the maximum tolerable error rate for the architecture’s operations.
  • Error accounting: The estimated operation errors include electronic rotations, hyperfine gates, photon absorption, and accumulated errors from repeated connection attempts.For pCZ, errors arise both during the connection attempts and during the final successful connection.
  • Target error rates: The architecture requires pI,M and pCZ to be both ≤0.1%, approximately an order of magnitude below the estimated threshold.The stricter target is motivated by the resource cost of operating near the threshold.
  • Parameter-set scope: The supplied parameter set satisfies the threshold condition but is not necessarily optimal, and the error expressions upper bound total operation errors.A detailed simulation is required to find optimal physical parameters across the full parameter space.
  • Expected performance: Measurement and initialization errors are below 0.1%, while the CPHASE error rate is above 0.1% but below the 0.73% threshold.The primary cause identified for the higher CPHASE error is the value of P2.
  • Error reduction: Reducing P2 could bring all error rates below the 0.1% target under the stated implementation assumptions.Suggested routes include tuning cooperativity and timing photon measurements so decay occurs near the 2π hyperfine-coupling point.

1. Expected performance

The architecture’s performance depends on connection success rates and operating mode, while topological error correction determines the resources and speed of logical operations.

  • ≈16 connection attempts at p_c = 6.25% require 3.5 µs in asynchronous mode, compared with ≈10^7 attempts and ≈22 µs synchronously.Asynchronous operation uses the average number of attempts, whereas synchronous operation waits until 99.9% of connections are established.
  • ≈21 µs per temporal layer and 42 µs per unit cell are achieved asynchronously, versus ≈132 µs and 264 µs synchronously.The circuit takes six steps to construct a temporal layer of the topological cluster state.
  • p_L ≤ 10^-18 requires code distance d ≥ 32, yielding 9,841 physical qubits and logical CNOT times of 3.4 ms asynchronously or 21.1 ms synchronously.The estimate assumes an average physical error rate p = 0.1% and a threshold error rate p_th ≈ 0.73%.
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