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Reducing the quantum computing overhead with complex gate distillation

Guillaume Duclos-Cianci, David Poulin

arXiv:1403.5280v1quant-ph

TL;DR

The paper addresses overhead caused by sparse distillation accuracies, costly components, and low-yield implementations. It develops recursive distillation with optimized schedules, higher-rate codes, and controlled-SWAP-based gadgets. The reported results include up to 3-fold improvement over m = 1, while the analysis notes assumptions that may understate the improvement.

  • Problem

    Standard distillation produces sparse achievable accuracies, forcing unnecessarily accurate and costly states during synthesis and recursive distillation.

    Source: Appendix A · p. 6

  • Method

    The scheme combines recursive magic-state distillation with optimized schedules, higher-rate code families, and controlled-SWAP gadgets.

    Source: Appendix A · p. 6

  • Results

    Up to 3-fold improvement over m = 1 is demonstrated by optimizing the code parameter at each distillation step.

    Source: Appendix A · p. 6

  • Takeaways & Limitations

    Higher-rate codes can both fine-tune distilled accuracies and reduce overhead within the recursive protocol.

    Source: Appendix A · p. 6

    The improvement calculation assumes access to lower-level states of arbitrary accuracy and uses a leading-order expansion.

    Source: Appendix A · p. 6

Abstract

from arXiv · show

In leading fault-tolerant quantum computing schemes, accurate transformation are obtained by a two-stage process. In a first stage, a discrete, universal set of fault-tolerant operations is obtained by error-correcting noisy transformations and distilling resource states. In a second stage, arbitrary transformations are synthesized to desired accuracy by combining elements of this set into a circuit. Here, we present a scheme which merges these two stages into a single one, directly distilling complex transformations. We find that our scheme can reduce the total overhead to realize certain gates by up to a few orders of magnitude. In contrast to other schemes, this efficient gate synthesis does not require computationally intensive compilation algorithms, and a straightforward generalization of our scheme circumvents compilation and synthesis altogether.

Appendix A: Further improvements

The appendix develops several ways to reduce the protocol’s compilation and distillation overhead, including optimized schedules, higher-yield implementations, and enlarged code families. These changes can improve yield and accuracy control, but may trade noise suppression against rate.

Appendix A: Further improvements

  • Cost-aware distillation schedules can reduce overhead by allowing costly components to contribute more to the final error.The baseline rule equalizes every noise contribution despite differing component costs.Source: Appendix A · p. 6
  • Direct implementation of Λ(SWAP) raises the rate from 1/8 to 1/7 for k > 3, but removes the code’s additional noise suppression.The benefit is therefore unclear early in the schedule, when noise is relatively high.Source: Appendix A · p. 6
  • Replacing the 4-qubit code with a [[2m + 2, 2m, 2]] family yields rate m/4(m + 1), asymptotically doubling the yield.The recursive protocol can amplify this yield improvement, while larger m increases opportunities for accumulated errors.Source: Appendix A · p. 6
  • The enlarged codes detect only one error, so detection scales as mϵ and harmful undetected errors as m2ϵ2.This worsens the m = 1 behavior but provides additional flexibility in scheduling.Source: Appendix A · p. 6
  • Up to 3-fold improvement over m = 1 is obtained by choosing the optimal code parameter at each distillation step.The enlarged code family fine-tunes otherwise sparse achievable accuracies while improving the rate.Source: Appendix A · p. 6
  • The reported improvement calculation uses a leading-order expansion and assumes arbitrary-accuracy lower-level states, potentially understating the gain from fine accuracy tuning.A more complete calculation is expected to produce an even larger improvement over m = 1.Source: Appendix A · p. 6

Appendix B: Error Analysis

This appendix provides calculation and simulation details for the circuits in Figure 2c).

Appendix B: Error Analysis

  • The appendix gives calculation details for the circuits of Figure 2c).Source: Appendix B · p. 6
  • The appendix gives simulation details for the circuits of Figure 2c).Source: Appendix B · p. 6
  • Figure 2c) is the stated scope of these appendix calculations and simulations.Source: Appendix B · p. 6

1. Imperfect Wk

The imperfect Wk analysis tracks how faulty resource-state injections propagate through the phase-gadget circuit. Measurement-dependent corrections combine errors while preserving their Y-type structure.

1. Imperfect Wk

  • Figure 4 compares m = 1 with optimized higher-rate codes for RY(θ), showing a factor improvement of ∼3.2 at ϵ = 10−13.The plotted regime uses θ < 8δ and relative accuracy 1/8.Source: Figure 4 · p. 7
  • Faulty |Yk−1⟩ injection makes the applied rotation depend on the measurement outcome, requiring a further RY(θk−2) correction after the negative outcome.The correction uses another faulty resource state, so errors propagate and combine across injections.Source: §1 · p. 7
  • Errors remain Y-type through successive injections, and cancellation requires matching error placements across the sequence.The analysis represents these combinations with an error-amplitude vector and a product operation.Source: §1 · p. 7
  • The imperfect application of RY(θk−1) yields an error vector for Wk, with rotation and measurement-outcome cases occurring with probabilities 1/2k−3 and 1/2k−i.Source: §1 · p. 7
  • The error analysis assigns probabilities to correction paths according to their measurement outcomes, including 1/2k−3 for applying all rotations and 1/2k−i for a specified stopping outcome.Source: §1 · p. 7

2. Λ(Swap) gadget

The Λ(SWAP) gadget is modeled by propagating Y errors through its encoded circuit and reducing them to effective stochastic maps. The resulting map also determines the probability of obtaining the trivial syndrome.

2. Λ(SWAP) gadget

  • Each of the eight RY(θ3) rotations in Λ(SWAP) is modeled as a Y-Pauli channel with strength γ after twirling the |Y3⟩ resource states.The twirling is possible because W3 = H is Clifford, making the resource-state error diagonal.Source: §2 · p. 7
  • Errors are pushed through the gadget and projected onto the trivial syndrome to keep the simulation on three qubits.Source: §2 · p. 7
  • Propagating each single-qubit Y error through the decoding circuit yields an effective stochastic map Γ.Γ is not trace preserving as defined, because it retains the syndrome-selection process.Source: §2 · p. 8
  • The probability of measuring the trivial syndrome in the first or second Λ(SWAP) gadget is computed as Tr(Γ(ρ(1,2))).Source: §2 · p. 8

3. Error Calculation and Simulation Details

The calculation propagates distilled-state errors and costs through post-selection, initialization, and recursively bootstrapped distillation rounds. The resulting costs account for producing two copies and for the probability that lower-level states are needed.

Error propagation

  • Post-selection produces the single-qubit output state, from which the output error parameters ϵout and ∆out are extracted.These quantities define the inputs to the cost calculation for a distillation round.Source: §3 · p. 8

Cost calculation

  • The cost of applying Wk sums the costs of states with 3 ≤k′ < k, weighted by their probability of being needed.The 1/2 prefactor reflects distilling two copies of ρk.Source: §3 · p. 8

Initialization

  • For 3 ≤k ≤8, the simulation starts from a state with 1% error and unit cost, while k ≥9 starts from |0⟩ with zero cost.For k ≥9, the resulting error parameters are ϵk,0 = sin^2(θk/2) and ∆k,0 = sin(θk)/2, with purely off-diagonal error.Source: §3 · p. 8

Recursive distillation

  • Each distillation round uses two copies of the input state to produce two improved copies, updating the error parameters and cost recursively.The procedure bootstraps from the k = 3 scheme to higher resource states.Source: §3 · p. 8

Appendix C: Euler angle decomposition

The appendix decomposes a small-angle rotation about an arbitrary Bloch-sphere axis into Z, Y, and X rotations. By equating axis-angle and Euler representations, it bounds every component angle by 2θ.

Decomposition setup

  • A small-angle rotation R̂n(θ) is decomposed as RZ(α)RY(β)RX(γ), with each component angle bounded in magnitude by 2θ.The decomposition applies after choosing the axis orientation so that 0 ≤θ ≤π/8.Source: Appendix C · pp. 8–9

Axis-angle representation

  • The axis-angle rotation matrix is written in terms of nx, ny, nz, θ, and trigonometric factors.Its entries provide the quantities matched against the Euler-angle matrix.Source: Appendix C · p. 8

Euler representation

  • The same rotation is expressed through the Euler-angle matrix for the sequence RZ(α)RY(β)RX(γ).Equating the two representations yields bounds on the three Euler angles.Source: Appendix C · p. 9

Angle bounds

  • The (3,1) matrix element implies |β| ≤2θ, while the (2,1) element gives |sin α| ≤(1 −cos θ + sin θ)/|cos β| and hence |α| ≤2θ.The bound on γ follows analogously from matrix element (3,2).Source: Appendix C · p. 9
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