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Distributed Quantum Computation Based-on Small Quantum Registers

Liang Jiang, Jacob M. Taylor, Anders S. Sørensen, Mikhail D. Lukin

arXiv:0709.4539v2quant-ph

TL;DR

The paper addresses how distributed quantum computation can remain robust when local registers are small and initialization, measurement, and entanglement generation are noisy. It develops a five-qubit register-based hybrid scheme using heralded optical entanglement, repeated QND measurement, entanglement pumping, and Markov-chain analysis. The scheme reports reasonable time overhead, high-fidelity non-local operations, and applicability to ion-trap and NV-center implementations.

  • Problem

    The paper asks what minimal local resources and time overhead are needed for robust distributed quantum computation with imperfect initialization, measurement, and entanglement generation.

  • Method

    The paper uses five-qubit registers with high-fidelity local unitaries, probabilistic heralded optical links, repeated QND measurements, entanglement purification, and Markov-chain models.

  • Results

    The scheme achieves deterministic non-local coupling gates with high fidelity and reasonable time overhead despite initialization, measurement, and entanglement-generation error probabilities near 5%.

  • Takeaways & Limitations

    Five or fewer qubits per register can support a robust hybrid architecture applicable to ion traps and NV centers, while probabilistic entanglement generation suffices for deterministic distributed gates.

  • Takeaways & Limitations

    The analysis must account for finite memory lifetime because purification increases operation time and can eventually make memory lifetime the limiting constraint.

Abstract

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We describe and analyze an efficient register-based hybrid quantum computation scheme. Our scheme is based on probabilistic, heralded optical connection among local five-qubit quantum registers. We assume high fidelity local unitary operations within each register, but the error probability for initialization, measurement, and entanglement generation can be very high (~5%). We demonstrate that with a reasonable time overhead our scheme can achieve deterministic non-local coupling gates between arbitrary two registers with very high fidelity, limited only by the imperfections from the local unitary operation. We estimate the clock cycle and the effective error probability for implementation of quantum registers with ion-traps or nitrogen-vacancy (NV) centers. Our new scheme capitalizes on a new efficient two-level pumping scheme that in principle can create Bell pairs with arbitrarily high fidelity. We introduce a Markov chain model to study the stochastic process of entanglement pumping and map it to a deterministic process. Finally we discuss requirements for achieving fault-tolerant operation with our register-based hybrid scheme, and also present an alternative approach to fault-tolerant preparation of GHZ states.

I. INTRODUCTION

The paper proposes distributed quantum computation using small local registers connected by probabilistic, heralded optical entanglement rather than direct state transfer. With five or fewer qubits per register, robust operations address noisy initialization, measurement, and entanglement generation while retaining high-fidelity local control.

  • I. INTRODUCTION: Hybrid architectures combine precise local control in small registers with optical long-range coupling to scale quantum computation.The approach is motivated by the difficulty of manipulating large multiqubit systems and the advantages of optical connections over large distances.
  • I. INTRODUCTION: Lossy optical channels favor heralded entanglement generation over direct state transfer because loss reduces success probability rather than silently corrupting accepted events.Polarization and wave-packet distortions can still reduce the fidelity of heralded entanglement, motivating purification.
  • I. INTRODUCTION: The paper asks how few local resources can support robust entanglement generation and what time overhead is required under imperfect operations.It also considers whether robustness can extend to initialization and measurement errors.
  • II. QUANTUM REGISTER AND EXPERIMENTAL IMPLEMENTATIONS: The proposed register uses one communication qubit, one storage qubit, and auxiliary qubits for purification and error correction, with high-fidelity local unitaries as a critical requirement.The paper focuses on ion traps and NV centers as candidate implementations.
  • III. UNIVERSAL QUANTUM COMPUTATION WITH TWO-QUBIT REGISTERS: FUNDAMENTALS: Probabilistic heralded entanglement can support deterministic non-local coupling gates, including controlled-U operations, without requiring deterministic entanglement generation.A Bell pair enables gate teleportation, and one Bell pair is consumed for a non-local controlled-U gate.
  • IV. ERRORS AND IMPERFECTIONS: The scheme uses up to three additional auxiliary qubits to suppress initialization, measurement, and entanglement-generation errors through repeated QND measurement and entanglement purification.The authors report reduced measurement errors, higher fidelity, and more efficient purification than the earlier protocol.
  • IV. ERRORS AND IMPERFECTIONS: Entanglement pumping remains probabilistic, so the paper analyzes time overhead and failure probability within a fixed computation clock cycle.A Markov-chain model incorporates stochastic entanglement-generation attempts into the purification process.

V. ROBUST MEASUREMENT & INITIALIZATION

The paper develops robust measurement, initialization, and two-level entanglement pumping for distributed quantum registers with noisy physical operations. Repeated QND measurement suppresses measurement errors, while pumping produces high-fidelity Bell pairs whose residual infidelity is ultimately limited by local-operation imperfections.

  • Robust measurement and initialization: A majority vote over 2m + 1 consecutive QND readouts provides robust measurement and supports measurement-based initialization.The QND circuit uses repeated CNOT operations and exploits commutation with the measured Z observable.
  • Robust measurement and initialization: With pI = pM = 5%, the effective measurement error can reach εM ≈ 8 × 10^-4 for m*=6 or εM ≈ 12 × 10^-6 for m*=10 when pL = 10^-4 or 10^-6, respectively.The paper also notes ion-trap measurements with εM as low as 6 × 10^-4.
  • Robust non-local coupling: The non-local coupling-gate error is dominated by Bell-pair infidelity when pL and εM are much smaller than 1 − F.This motivates generating high-fidelity Bell pairs before implementing non-local gates.
  • Two-level entanglement pumping: Two-level pumping first purifies bit errors with raw Bell pairs, then purifies phase errors using the bit-error-purified pairs.Unsuccessful attempts discard the stored pair and restart the pumping process.
  • Fidelity of entanglement pumping: In the ideal-operation limit pL, εM → 0, the purified-pair infidelity can approach zero, whereas standard pumping retains an infidelity larger than (1 − F)^2 / 9.The new scheme therefore reduces the qubit-resource burden for very small local-operation and measurement errors.
  • Fidelity of entanglement pumping: For dephasing-dominated pairs, one-level pumping can produce a very high-fidelity pair after np = 3 successful pumping steps; for depolarizing errors, two-level pumping is required.The minimum achievable infidelity is limited by εM and pL under the optimized parameters considered.

VII. MARKOV CHAIN MODEL

The paper models one- and two-level entanglement pumping as finite-state Markov chains, using state-transition probabilities to calculate success and failure after a finite number of raw Bell pairs. For sufficiently large Ntot, the failure probability decreases exponentially and can be efficiently suppressed.

  • One-level pumping: One-level pumping uses an initial state, an unpurified-pair state, intermediate purified-pair states, and a final self-trapped state.Success advances the chain, while failure returns it to state 0.
  • Markov-chain calculation: The probability vector evolves by repeated multiplication with a transition matrix from the initial distribution, yielding the full state distribution after Ntot attempts.The final state probability gives the success probability, and its complement gives failure probability.
  • Two-level pumping: The two-level pumping failure probability is calculated with a Markov chain containing (nb + 1)(np + 1) + 1 states.nb and np denote the pumping steps used for the two levels.
  • Results: For (nb, np) = (2, 3) and (3, 4), the failure probability decreases exponentially toward zero as Ntot becomes sufficiently large.A reasonably large number of raw Bell pairs can therefore suppress failure efficiently.

B. Total error probability & average infidelity

The paper combines pumping failure probability with purified-pair infidelity to define total error probability, and separately evaluates average infidelity when partially purified pairs are retained. Both optimized measures approach the same minimum as the resource overhead increases.

  • Error metrics: Total error probability is approximated as the sum of pumping failure probability and purified Bell-pair infidelity.The minimum value is the minimal infidelity achievable by entanglement purification.
  • Conservative estimate: The total-error estimate is conservative because partially purified Bell pairs are sometimes created but treated as having zero fidelity when they are not the targeted pair.Average infidelity incorporates these partially purified outputs instead.
  • Error metrics: Average infidelity is a weighted average over the Markov-chain output states and also accounts for cases where no partially purified pair remains.Those cases are conservatively assigned infidelity 1/2, and average infidelity is generally smaller than total error probability.
  • Optimization: Both optimized total error probability and optimized average infidelity asymptotically approach the same minimum value as Ntot increases.The control parameters (nb, np) can be optimized for these measures.
  • Dephasing-error regime: One-level pumping is sufficient for Bell pairs dominated by dephasing error, with nb = 0 reducing resource requirements.The optimized error measures for this case are plotted as functions of Ntot.

C. Total time for robust entanglement generation

The paper maps stochastic entanglement generation to a deterministic time-and-error model using the raw-pair overhead Ntot. Tens to hundreds of raw Bell pairs can provide high success probability, while non-post-selective pumping can substantially reduce overhead in a low-fidelity regime.

  • Time model: The robust entanglement-generation time is proportional to the average number of raw Bell pairs generated, with additional local-operation and measurement contributions.The stochastic generation process can be incorporated as a two-state sub-level in the Markov chain.
  • Time model: For Ntot > 20, the relative deviation in the number of generated Bell pairs is approximately Ntot^-1/2 and has only minor influence, so the average can be replaced by Ntot.This approximation treats the raw-pair overhead deterministically.
  • Quality-overhead trade-off: The optimized error measures approach their asymptotic minimum, so increasing Ntot eventually yields little further quality improvement.This establishes the trade-off between gate quality and time overhead.
  • Quality-overhead trade-off: With pI = pM = 5% and initial fidelity F0 > 0.95, total error is mostly limited by pL, with an overhead factor of about 10 largely insensitive to F0.The pumping process requires tens or hundreds of raw Bell pairs for very high success probability.
  • Quality-overhead trade-off: The total-error-based Ntot estimate is approximately 1.2–2 times larger than the average-infidelity-based estimate because it is more conservative.Both estimates are obtained from the corresponding error measures.
  • NPS pumping: The Markov-chain approximation is optimistic when local operational errors are finite because the score alone no longer specifies intermediate Bell-pair states.It remains useful when local operational errors are small compared with intermediate-pair infidelity.
  • NPS pumping: For F < 0.9 and pL < 10^-4, non-post-selective pumping improves the raw-pair overhead by more than a factor of 3 relative to post-selective pumping.The NPS scheme avoids restarting after intermediate failures by reducing the chain score instead.

A. Time and error in the theoretical model

The model relates clock-cycle time and effective error probability to optical, local-operation, initialization, measurement, and entanglement-generation imperfections. Numerical estimates indicate reasonable overheads for ion-trap and NV-center implementations, while finite memory time constrains achievable operation rates.

  • Theoretical model: The clock cycle and effective error probability are expressed as functions of the relevant optical and operation imperfection parameters.The model combines initialization, measurement, entanglement-generation, radiative-decay, detection-efficiency, fidelity, and local-operation parameters.
  • Theoretical model: The photon-emission-to-local-operation time ratio is typically below 0.01 for ion-trap and NV-center systems.This dimensionless ratio compares the time to emit one photon with the time for a local unitary operation.
  • Practical estimates: The clock cycle becomes longer with lower Bell-pair fidelity or higher initialization and measurement errors, but decreases when errors change from depolarizing to dephasing.The table evaluates pL values from 10−3 to 10−6 and assumes pM = pI = 1 − F.
  • Memory constraint: Finite memory time ultimately limits purification because longer operations accumulate memory errors, despite purified gate fidelity being limited only by local-operation imperfections.For fault tolerance, the memory error per clock cycle is approximately tC/tmem and should be small, such as 10−4.

IX. APPROACHES TO FAULT TOLERANCE

The scheme supports fault-tolerant computation through non-local gates and an alternative partial-Bell-measurement construction for GHZ states. The PBM approach reduces register requirements while preserving fault-tolerant error behavior and scales efficiently to larger GHZ states.

  • Fault-tolerant computation: The entanglement-based approach can implement gates between arbitrary registers, with quantum error correction available when gate errors are sufficiently small.The paper reports effective errors of approximately 2.7 × 10−3 and 1.7 × 10−5 for coding regimes discussed later.
  • Fault-tolerant computation: The scheme estimates 20 registers per logical qubit for K = 10^4 logical qubits and Q = 10^6 logical operations at γ ≈ 1.7 × 10−5.The estimate assumes tC/tmem ≈ γ, with tmem ≈ 10 s and tC ≈ 162 µs.
  • GHZ preparation: Partial Bell measurements enable fault-tolerant four-qubit GHZ preparation with four registers instead of the eight required by the conventional circuit.A redundant PBM detects bit errors from earlier PBMs, while PBMs propagate neither bit nor phase errors.
  • GHZ preparation: PBMs project storage qubits into Φ± or Ψ± subspaces according to whether measurement outcomes agree or differ.For differing outcomes, one storage qubit may be flipped to reach the Φ± subspace.
  • GHZ preparation: A 2^n-qubit GHZ state can be prepared with 2^n registers; for n ≥ 3, preparation takes three clock cycles with error probability approximately 3p/2 per register.PBMs acting on different registers can be performed in parallel.
  • Conclusion: The hybrid scheme requires five or fewer qubits per register and uses a Markov-chain model to analyze robust measurement and entanglement-generation overheads.The conclusion also identifies GHZ preparation via PBMs as an example for fault-tolerant quantum computation.
  • Entanglement purification: The two-level pumping scheme can produce Bell pairs with arbitrarily high fidelity for sufficiently large pumping parameters under the stated assumptions.The contour result assumes depolarizing error with initial fidelity F = 0.95 and vanishing measurement and local-channel errors.

APPENDIX A: BIT-PHASE TWO-LEVEL PUMPING SCHEME

The appendix establishes that the bit-phase two-level entanglement-pumping scheme can asymptotically purify Bell pairs toward unit fidelity under perfect local operations. Its analysis represents Bell-diagonal states with a compact fidelity vector and tracks pumping recursively.

  • Appendix A: With perfect local operations, bit-phase two-level pumping can create Bell pairs with fidelity arbitrarily close to unity.The appendix provides a rigorous proof supported by numerical contours.
  • Appendix A: Bell-diagonal states are represented by the fidelity vector F⃗ = (a, b, c, d), whose coefficients are non-negative and sum to one.The representation remains valid after purification, so four coefficients suffice for each state.
  • Appendix A: The pumping analysis computes each step’s success probability and updated fidelity vector recursively from the pumped and pumping states.For bit-error pumping, the success probability is given explicitly in Eq. (A2).

1. First level pumping

The first-level pumping analysis parameterizes Bell-state imperfections and derives recursive bounds for bit- and phase-error purification. These bounds show exponentially decreasing residual error under the stated Werner-state and perfect-operation assumptions.

  • First level pumping: The first pumping level parameterizes the fidelity vector to analyze purification against bit errors.The same framework is extended to phase-error pumping through its corresponding success-probability relation.
  • First level pumping: For the initial bound, ηn remains below η0 = α/2 < 1/6 throughout the pumping sequence.This bound is used in subsequent estimates of the pumping success probability.
  • First level pumping: The residual parameter ηn decreases exponentially with pumping step n, making the success probability approach 1 − α for sufficiently large n.The asymptotic expression is pn ≈ 1 − α − O(ηn).
  • Assumptions: The analysis assumes an initial Werner state with F0 > 1/2 and perfect local operations for the purification bounds.The Werner-state assumption is used as a worst-case representation with the same fidelity.

2. Second level pumping

The second-level bit-phase pumping analysis bounds fidelity-vector components across pumping stages and selects pumping-step counts to approach unit fidelity. The resulting scheme can achieve arbitrarily high Bell-pair fidelity in principle.

  • Error-component behavior: The second-level pumping further purifies the erroneous |Ψ−⟩ admixture, while the |Ψ+⟩ admixture may increase but remains upper-bounded.These two error components therefore have different behaviors during second-level pumping.
  • Choice of pumping steps: The analysis uses inequalities on fidelity-vector elements to choose first- and second-level pumping steps that drive the fidelity arbitrarily close to unity.The selected step counts are denoted nb and np.
  • Achievable fidelity: The bit-phase two-level pumping scheme allows F = 1 to be approached with arbitrary good precision.This is the stated achievable-fidelity conclusion for the chosen pumping parameters.

APPENDIX B: MARKOV CHAIN MODEL FOR TWO-LEVEL PUMPING

The Markov-chain model represents two-level entanglement pumping as stochastic transitions among states that track progress through both pumping levels. Contracting a non-consuming transition yields a finite transition system whose probability distribution evolves through repeated attempts.

  • State representation: Two labels track the intermediate state because two entangled pairs are stored during two-level entanglement pumping.The states record progress at the first and second pumping levels.
  • Transition rules: Success and failure transitions are governed by qj at the first level and Qk at the second level.First-level failures reset the first-level progress, while second-level failures return to the initial state.
  • Probability evolution: A probability vector P evolves from attempt t to t + 1 through a transition matrix describing the contracted state space.The model therefore maps repeated stochastic pumping attempts to probability-distribution updates.

APPENDIX C: FAULT-TOLERANT PREPARATION OF 2n-QUBIT GHZ STATE

The alternative GHZ-state construction prepares a 2^n-qubit GHZ state by applying pairwise PBM operations among labeled registers in parallel. The stated construction uses three clock cycles.

  • Preparation schedule: A 2^n-qubit GHZ state with n ≥ 3 can be prepared in 3 clock cycles using parallel PBM operations on prescribed register pairs.The three cycles connect different register-pair groupings in succession.
  • Target state: The construction targets the specific GHZ state |00 · · · 0⟩+.The passage introduces this target state after describing the parallel preparation schedule.
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