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Magic-state distillation with the four-qubit code

Adam M. Meier, Bryan Eastin, Emanuel Knill

arXiv:1204.4221v1quant-ph

TL;DR

The paper addresses the resource cost of producing high-quality |H⟩ states for quantum computation. It introduces a four-qubit-code routine and finds that combining it with the 15-to-1 routine can substantially lower production cost, while output errors from a single 10-to-2 instance may be correlated.

  • Problem

    The paper considers the cost of obtaining high-quality resource states for useful quantum computations, especially when input-state error probabilities range from 0.01 to 10^-5.

    Source: §IV · p. 5, §V · p. 7

  • Method

    The routine encodes resource states in a four-qubit error-detecting code, verifies them with an encoded measurement, and decodes accepted outputs.

    Source: §II · p. 3

  • Results

    At p = 0.01, a goal error near 10^-5 has production cost 27.9 for two 10-to-2 rounds versus 261.7 for two 15-to-1 rounds, a factor-of-9.4 improvement.

    Source: §V · p. 8

  • Takeaways & Limitations

    Combining the 10-to-2 routine with the 15-to-1 routine can reduce the number of faulty |H⟩ states required by up to an order of magnitude.

    Source: §VI · pp. 8–9

    Outputs from one 10-to-2 instance can have correlated errors, so multi-round sequences cannot generally assume independent output states.

    Source: §IV · p. 6

Abstract

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The distillation of magic states is an often-cited technique for enabling universal quantum computing once the error probability for a special subset of gates has been made negligible by other means. We present a routine for magic-state distillation that reduces the required overhead for a range of parameters of practical interest. Each iteration of the routine uses a four-qubit error-detecting code to distill the +1 eigenstate of the Hadamard gate at a cost of ten input states per two improved output states. Use of this routine in combination with the 15-to-1 distillation routine described by Bravyi and Kitaev allows for further improvements in overhead.

I. BASIC CONCEPTS AND NOTATION

The section establishes the Clifford-operation setting and defines magic-state distillation as a way to turn faulty resource states into fewer, improved states. It also specifies the paper’s error model and notation for the |H⟩ resource state.

I. BASIC CONCEPTS AND NOTATION

  • |H⟩ is the +1 Hadamard eigenstate and a non-stabilizer resource state that Clifford operations alone cannot prepare.Together with Clifford operations, |H⟩ can implement a non-Clifford gate.Source: §I · p. 2
  • Distillation projects faulty magic states into a suitable subspace; successful projections yield higher-fidelity states, while detected failures are discarded.The analysis assumes Clifford operations can be implemented perfectly.Source: §I · p. 2
  • The paper assumes resource states are twirled with H before use, so faulty |H⟩ states are modeled as stochastic Y errors.The twirling operation produces a probabilistic mixture of |H⟩ and |−H⟩ states.Source: §I · pp. 2–3
  • The paper labels distillation routines by their input-to-output state counts, so an m-to-n routine consumes m resource states and produces n.Source: §I · p. 3

II. 10-TO-2 DISTILLATION ROUTINE

The routine encodes faulty |H⟩ states in the four-qubit error-detecting code, verifies them with an encoded Hadamard measurement, and decodes accepted outputs. Its two encoded outputs are retained only when neither measurement nor syndrome indicates an error.

II. 10-TO-2 DISTILLATION ROUTINE

  • The routine encodes resource states, verifies them with an encoded measurement, and decodes to obtain improved states when no errors are indicated.Outputs are discarded if either the encoded measurement or the C4 syndrome indicates an error.Source: §II · pp. 3–4
  • The encoded Hadamard measurement tests whether the pair occupies the subspace spanned by |H⟩|−H⟩ and |−H⟩|H⟩, detecting an error on either state.Source: §II · p. 3
  • The C4 code detects any single-qubit error because every one-qubit Pauli operator anticommutes with at least one stabilizer generator.Source: §II · p. 3
  • The C4 code’s transversal Hadamard implements logical Hadamards on both encoded qubits, enabling the encoded measurement with four CH gates and eight resource states.The encoded circuit reduces the CH-gate requirement compared with the unencoded construction.Source: §II · p. 4

III. ANALYSIS

The analysis classifies input errors by whether they are detected or cause logical errors, then derives acceptance and conditional output-error probabilities. Any single error is detected, while undetected errors occur at order p^2 and the two outputs’ errors are positively correlated.

III. ANALYSIS

  • Data-state Y errors flip the encoded measurement, while a single gate-state error is detected by the syndrome measurement.Circuit identities concentrate errors into regions where undetected patterns can be classified by their logical effect.Source: §III · p. 4, Figure 4 · p. 5
  • The routine accepts with probability a(p), whose exact accounting follows from enumerating error patterns that pass detection.The analysis assumes independent input states with the same error probability p and perfect Clifford operations.Source: §III · p. 4
  • Any single error is detected, so the probability of an accepted output error is of order p^2.The conditional marginal error probability is the quality measure for each output state.Source: §III · pp. 4–5
  • The two outputs’ errors are positively correlated: the conditional probability of at least one error is at most 2e(p) − e(p)^2, and the probability of errors on both is order p^2.The marginal error probability e(p) is the same for either output state.Source: §III · pp. 4–5

IV. DISTILLATION SEQUENCES

The section formulates multi-round distillation to produce many resource states below a target marginal error probability while minimizing input cost. It accounts for correlations between outputs by arranging rounds so each instance uses at most one output from any earlier instance.

IV. DISTILLATION SEQUENCES

  • To reach very low error probabilities, multiple rounds group outputs from the preceding round into blocks for further distillation.Outputs from one routine may have correlated errors.Source: §IV · pp. 5–6
  • The sequence of rounds is chosen to minimize input states per output while meeting the target marginal error probability.The cost is considered asymptotically when many resource states are prepared.Source: §IV · pp. 5–6
  • A 10-to-2 routine can produce correlated outputs, so multi-round instances must not reuse more than one output from any preceding instance.This strategy permits asymptotic cost calculations as if outputs were independent, without increasing asymptotic cost.Source: §IV · p. 6
  • With output symmetrization, blocks formed from a block of K independent inputs have independent states within each output block, each with error probability e(p).Each block contains a(p)⌊K/m⌋ states on average.Source: §IV · p. 6
  • The Fig. 5 curves compare conditional marginal error probabilities for the 10-to-2 and 15-to-1 routines; each threshold is their first intersection with the no-distillation line.Source: Figure 5 · p. 6

V. COMPARATIVE PERFORMANCE

The section compares the 10-to-2 and 15-to-1 routines across practical input-error rates, including sequences that combine them. It finds that 10-to-2 can substantially reduce production cost, while 15-to-1 has a higher threshold and greater per-round error improvement.

V. COMPARATIVE PERFORMANCE

  • The 10-to-2 threshold is 0.089, below the 15-to-1 threshold of 0.141, and both are described as adequate for the error regime of interest.Threshold is the input error probability below which the routine can distill resource states.Source: §V · p. 7
  • In the small-input-error limit, 10-to-2 has an asymptotic output-error exponent of about 0.43 versus 0.4 for 15-to-1, but integral round lengths create practical step discontinuities.The smooth asymptotic functions can therefore mislead practical comparisons.Source: §V · p. 7
  • For a given input error and goal error, the nearest curve above the point in Fig. 6 identifies the best sequence of 10-to-2 and/or 15-to-1 rounds.Each curve labels a sequence, and the region beneath it marks where that strategy is preferred.Source: §V · p. 7, Figure 6 · p. 7
  • The 10-to-2 routine has lower cost, whereas 15-to-1 yields greater error-probability improvement per round.Numerically, when combining the routines, any 15-to-1 rounds should come first; this is consistent with its higher threshold and suggested advantage at high input error.Source: §V · pp. 7–8
  • Near a goal error of 10^-5 from input error 0.01, two 10-to-2 rounds cost 27.9 versus 261.7 for two 15-to-1 rounds, a 9.4-fold cost improvement.Fig. 7 compares the best sequences against the best sequence using only 15-to-1; its gray regions indicate improvement from using 10-to-2.Source: Figure 7 · p. 8, §V · p. 8

VI. CONCLUSIONS

The conclusion presents magic-state distillation as a route to universal quantum computing from mediocre non-stabilizer states with high-quality Clifford operations. It reports a 10-to-2 |H⟩ routine and up to an order-of-magnitude reduction in input states when combined with earlier routines.

VI. CONCLUSIONS

  • Table I reports costs and output error probabilities at p = 0.01, with cost improvement measured against the shortest 15-to-1-only sequence achieving at least as good an output error probability.Source: Table I · p. 8
  • At least one high-quality magic state is required for indirect implementation of each non-Clifford gate, motivating reduced distillation costs.Source: §VI · p. 8
  • The new |H⟩ routine reduces error probability from p to O(p^2) while producing two outputs from ten inputs.Source: §VI · p. 8
  • Combining the routine with distillation routines from Refs. [2] [3] can reduce the faulty |H⟩ states needed for a target quality by up to an order of magnitude.Source: §VI · p. 9

Appendix

The appendix supplies circuit-identity details for circuits introduced in Fig. 2. These identities derive the encoded Hadamard gate and explain elimination of a pair of CH gates.

Appendix

  • The encoded Hadamard circuit applies H to the third physical qubit after decoding the second logical qubit, then re-encodes.Source: Figure 8 · p. 10
  • Commuting and cancelling CX pairs, decomposing CX into CZ and H gates, and using H^2 = I establish successive circuit equivalences.Source: Figure 8 · p. 10
  • The final equivalence uses ZX = iY and permits exchanging same-target CX and CZ gates by adding a CZ between their controls.Source: Figure 8 · p. 10
  • A pair of same-target CH gates can be eliminated by exchanging CH and CY gates with an added CZ between their controls, since H anticommutes with Y.Source: Figure 8 · p. 10
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