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Magic state distillation in all prime dimensions using quantum Reed-Muller codes

Earl T. Campbell, Hussain Anwar, Dan E. Browne

arXiv:1205.3104v2quant-ph

TL;DR

Magic state distillation for odd-prime-dimensional systems requires effective protocols beyond the qubit setting. The paper uses quantum Reed-Muller codes with transversal non-Clifford gates and develops qutrit and ququint schemes. Its five-dimensional protocol achieves a 36.3% depolarizing-noise threshold and superior yield to known qubit protocols, while higher-dimensional performance deteriorates beyond d = 5 and relevant fault-tolerance thresholds remain unknown.

  • Problem

    Magic state distillation protocols are needed for fault-tolerant schemes in odd prime dimensions, where higher-dimensional systems may admit codes without direct qubit analogues.

  • Method

    The paper constructs magic state distillation protocols using quantum Reed-Muller codes with transversal non-Clifford gates, including qutrit and ququint schemes.

  • Results

    The ququint protocol achieves a depolarizing-noise threshold of 36.3% and superior yield to all known qubit protocols, while the qutrit protocol remains competitive with qubit protocols.

  • Takeaways & Limitations

    For primes d ≥ 5, quantum Reed-Muller codes with only d−1 qudits can provide transversal non-Clifford gates, with the d = 5 protocol delivering practical gains.

  • Takeaways & Limitations

    Comparable fault-tolerance thresholds for higher-dimensional systems remain unknown, and the available qudit-threshold analysis is limited.

Abstract

from arXiv · show

We propose families of protocols for magic state distillation -- important components of fault tolerance schemes --- for systems of odd prime dimension. Our protocols utilize quantum Reed-Muller codes with transversal non-Clifford gates. We find that, in higher dimensions, small and effective codes can be used that have no direct analogue in qubit (two-dimensional) systems. We present several concrete protocols, including schemes for three-dimensional (qutrit) and five-dimensional (ququint) systems. The five-dimensional protocol is, by many measures, the best magic state distillation scheme yet discovered. It excels both in terms of error threshold with respect to depolarising noise (36.3%) and the efficiency measure know as "yield", where, for a large region of parameters, it outperforms its qubit counterpart by many orders of magnitude.

I. STABILIZER OPERATIONS AND THE MAGIC STATES MODEL

The magic states model treats stabilizer operations as protected but classically simulatable, so non-stabilizer magic states supply the missing resource for universal computation. In odd prime dimensions, the paper defines suitable M-type gates and Reed–Muller-code protocols that quadratically suppress noise.

  • Quantum Reed–Muller codes generalize the qubit 15-qubit protocol and provide transversal non-Clifford gates for magic-state distillation.
  • For qutrits, suitable M-type gates exist for m ≥2, while for prime d ≥5 they exist for m ≥1.
  • The protocols distill non-stabilizer eigenstates and use stabilizer operations to iteratively reduce noise quadratically.

A. CSS codes

The paper represents CSS stabilizer codes through classical vector spaces and logical Pauli operators, then identifies code properties that support magic-state distillation. A code of distance D yields error suppression proportional to ϵ^D.

  • CSS codes are specified by phase- and bit-flip stabilizer subgroups corresponding to classical vector spaces LZ and LX.
  • For a CSS code, the number of logical qudits is k = n − Dim(LZ) − Dim(LX).
  • Logical operators XL and ZL define the encoded qudit basis and obey the same conjugation relation as the physical X and Z operators.
  • A suitable distillation code requires logical transversality, with M ⊗n implementing the logical operator M†L.
  • Error suppression scales as ϵ′ ≤ Kϵ^D, and therefore a positive distillation threshold exists below which errors decrease iteratively.

C. The protocol

The protocol twirls noisy magic states, applies stabilizer measurements and outcome-dependent Clifford corrections, postselects, and decodes one qudit. Transversality reduces magic-state distillation to an X-basis error-correction problem.

  • It measures phase stabilizers, applies outcome-dependent Clifford corrections, measures bit-flip stabilizers, postselects, and decodes the surviving qudit.
  • The protocol can be iterated by feeding the decoded output state back as the next input.
  • The protocol applies CM-twirling before code projection to convert input states into a canonical form.
  • Transversality makes distillation of |M0⟩ equivalent to a simpler distillation problem in the X basis.
  • The output is diagonal in the M† basis, and cycling requires replacing CM by C†M on odd iterations.

D. Analyzing the iterative formulae

The analysis expresses output fidelities through code-dependent weight enumerators and studies both depolarizing and general noise. The code distance determines the leading error-suppression order and guarantees a threshold.

  • The iterative fidelity formula simplifies under depolarizing noise because the noise model depends on a single parameter and Hamming weights.
  • Taylor expansion shows that the leading error suppression has degree D in the rescaled noise parameter.
  • Since D ≥2, the protocols provide at least quadratic error suppression.
  • For arbitrary noise models, bounding all nonzero noise parameters by µ yields error terms controlled by the code distance D.
  • A valid threshold follows from ϵ∗ = K^−1/(D−1), although the general bound is loose and larger thresholds may exist.

E. Clifford correction

The protocol uses Clifford corrections to convert arbitrary stabilizer-measurement outcomes into the desired codespace outcome, increasing each round’s success probability. Concrete qutrit and ququint Reed–Muller codes supply the required transversal non-Clifford gates.

  • Clifford correction: Clifford correction maps arbitrary measurement outcomes to the desired projector outcome by applying a correction determined by the measured syndrome.The correction condition is ⟨k,u⟩ = ⟨w,Gu⟩ mod d; a canonical generator matrix provides a direct choice of w.
  • Clifford correction: The correction strategy significantly increases the success probability of each distillation round.Success approaches certainty in the limit of pure initial states.
  • Qutrit example: QRM3(2) is an eight-qutrit CSS code whose logical operators are ZL = Z[21] and XL = X[1].The code is transversal with respect to the canonical M3 non-Clifford gate.
  • Ququint example: QRM5(1) is a four-ququint CSS code with logical operators ZL = Z[41] and XL = X[1].It is transversal with respect to the canonical M5 non-Clifford gate.
  • Ququint example: For d = 5, phases that are multiples of ω can define a non-Clifford gate, enabling smaller codes with transversal non-Clifford operations.The same phase structure is Clifford for dimensions below d = 5.

B. Classical Reed-Muller codes

The paper reviews d-ary Reed–Muller codes as polynomially defined linear codes and identifies a symmetry property that supports magic-state constructions. A λ-function sums to zero modulo dm on every unshortened first-order codeword.

  • Code construction: RMd(1,m) consists of affine functions over Fd, with code length n = dm and dimension m + 1.First-order codes use degree-1, hence linear, polynomials; affine functions add a constant term.
  • Code construction: The family of linear maps from Fd^m onto Fd provides the underlying vector-space structure for these codes.The construction enumerates linear maps using coefficient vectors and points represented in base d.
  • λ-functions: A λ-function assigns one of d integer values according to the symbols appearing in a codeword.These functions are closely related to the non-Clifford gates used later.
  • λ-functions: For every λ-function Λ and every v in RMd(1,m), Λ(v) = 0 mod dm.The proof uses uniform multiplicities of field values across nontrivial affine codewords and invariance under changes of variables.
  • Code construction: Unshortened Reed–Muller codes have substantial affine symmetry, but the paper shortens them to remove enough symmetry for the protocol.The shortening step is introduced because the unshortened codes have too much symmetry for the intended application.

C. Shortened classical Reed-Muller codes

Shortening removes the constant-coordinate contribution from Reed–Muller codes, producing smaller linear codes used to construct quantum CSS codes. Their λ-function relation directly supports transversal gates and distance-2 distillation codes.

  • Code shortening: A shortened Reed–Muller code keeps unshortened codewords whose first coordinate is zero and deletes that coordinate.It has length n − 1 and can also be defined directly using linear maps rather than affine maps.
  • Code shortening: The shortened code has dimension m, one less than the corresponding unshortened code.The lost dimension corresponds to removing the constant term in the affine-function description.
  • λ-function relation: For a shortened codeword v, Λ(v ⊕ c1) = −λc mod dm.Appending the omitted coordinate produces an unshortened codeword, allowing Lemma 1 to establish this relation.
  • Quantum construction: The quantum code QRMd(m) is a CSS code over n = dm − 1 qudits of prime dimension d.Its stabilizer is built from the shortened code and its dual, with logical operators required to commute appropriately.
  • Quantum construction: The code distance is bounded by the smallest-weight logical phase or bit-flip operator, with QRMd(m) attaining the nontrivial lower bound of 2.A weight-1 logical error would require a qudit on which the X stabilizer acts trivially, which does not occur.

E. MacWilliams identities

MacWilliams identities simplify performance analysis by relating weight enumerators of the smaller stabilizer codes to their duals. The resulting Reed–Muller protocols have quadratic error suppression and polynomial resource cost, with QRM5(1) achieving the best yield scaling reported.

  • Weight enumerators: MacWilliams identities relate the weight enumerator of a code to that of its dual, simplifying depolarizing-noise calculations.The smaller LX and L′X codes make this relation especially useful because their duals are larger and more complex.
  • Error suppression: For protocols based on QRMd(m), the error suppression is quadratic for every odd prime d and every m.This contrasts with the qubit QRM2(4) protocol, whose reduction is cubic: ϵ′ ∼ 35ϵ^3.
  • Protocol performance: QRM5(1) has the best yield scaling among quantum Reed–Muller codes and retains that distinction against all presently known protocols.The paper compares yield in the limit of many iterations through the parameter γ* and the associated table.
  • Yield scaling: The expected resource cost increases only polynomially in ϵ_target^-1.This follows from quadratic error reduction and the yield analysis based on repeated distillation rounds.
  • Yield scaling: For odd-prime protocols, the distance is D = 2 and the yield-scaling parameter is γ* = log2(dm − 1).The parameter governs asymptotic resource efficiency, with smaller γ* indicating better yield scaling.

B. Depolarizing noise thresholds

The protocols achieve quadratic error suppression in odd prime dimensions, with QRM5(1) and QRM3(2) attaining strong depolarizing-noise thresholds. QRM3(2) uses few copies and has a broad distillable region, although its threshold and yield trade off against iteration size and noise level.

  • Protocol trade-offs: Increasing m raises the number of copies required per iteration but lowers the depolarizing threshold, making the smallest viable m advantageous.Larger m instead enlarges the set of states that the protocol can distill.
  • Threshold comparison: 36.3% is the reported depolarizing threshold for QRM5(1), exceeding the qubit Reed–Muller comparison and reflecting the advantage of odd prime dimensions.The paper attributes this advantage to smaller codes with transversal non-Clifford gates; QRM5(1) uses 4 ququints.
  • QRM3(2) performance: Quadratic suppression per iteration distinguishes QRM3(2) from the earlier qutrit protocol, which showed only linear suppression.QRM3(2) uses 8 copies per iteration and has success probability P ≥ 1/9 for all states.
  • QRM3(2) thresholds: 0.20015 is the general-noise threshold for QRM3(2), while 0.211001 is its depolarizing-noise threshold.Below the general threshold, output error is lower than input error for all noise orientations; a quadratic bound gives ϵ′ ≤ 5.03ϵ2.
  • Distillable region: The distillable qutrit region extends beyond the simple ϵ < ϵ∗ criterion, but an intermediate regime remains unresolved by this protocol and existing theorems.Some noise orientations tolerate greater noise than the general threshold, while ambiguous states are neither distilled by QRM3(2) nor ruled out as distillable.
  • Yield comparison: QRM3(2) yield exceeds BK by many orders of magnitude at higher input error, although BK has slightly better asymptotic scaling at very small target error.BK's yield vanishes near its threshold of approximately 0.1415, whereas QRM3(2) tolerates depolarization up to approximately 0.211.

D. Peformance of QRM5(1)

QRM5(1) is presented as a compact five-dimensional distillation protocol with strong depolarizing-noise protection and high yield, while state-injection analysis connects purified magic states to deterministic non-Clifford gates.

  • QRM5(1) protocol: QRM5(1) is the first protocol applied to magic-state distillation in five-dimensional systems.
  • Yield: QRM5(1) has the best known expected-yield scaling, with γ = 2, and numerics report resource savings of potentially many orders of magnitude across parameter regimes.The yield comparison is against the qubit protocol QRM2(4), or Bravyi and Kitaev.
  • Noise protection: 36.3% is QRM5(1)'s reported depolarizing-noise threshold, with threshold values also reported for generic noise.The passages report ϵ∗dep = 0.363122 and a generic-noise threshold of 0.31195.
  • Scope of analysis: For five-dimensional systems, analysis focuses on depolarized states because the full distillability region is difficult to represent visually.The depolarized case uses f0 = 1 − ϵ and fj≠0 = ϵ/4.
  • Error transformation: The protocol maps a depolarized input error ϵ to an output error ϵ′ given by a rational polynomial expression.The displayed expression has numerator ϵ2(96 − 160ϵ + 75ϵ2) and denominator 64 − 256ϵ + 480ϵ2 − 400ϵ3 + 125ϵ4.
  • State-injection: State-injection uses a magic state and a stabilizer operation to implement the desired unitary deterministically, including for noisy resources with bounded error.The construction first handles perfect magic states and then extends to imperfect states parameterized by ϵ.
  • State-injection: The injection transformation is initially random among d possibilities, but applying an inverse Clifford correction recovers the desired M unitary.The correction is described as (X^k)†C^k.
  • Universal quantum computing: Adding non-Clifford M and M† gates to Clifford operations generates a set dense in the special unitary group, enabling efficient approximation through Solovay–Kitaev.

VII. DISCUSSION

The paper generalizes magic state distillation with quantum Reed–Muller codes to all prime dimensions, identifying especially strong qutrit and ququint protocols. The ququint protocol combines a four-ququint code with strong depolarizing-noise thresholds and yield, while broader comparisons and fault-tolerance implications remain qualified.

  • Generalization: Quantum Reed–Muller codes generalize the qubit magic-state distillation approach to all prime dimensions.The construction prepares highly purified non-stabilizer states using ideal stabilizer operations, which can support universal computation through state injection.
  • Ququint protocol: The 4-ququint code is, to the authors’ knowledge, the smallest non-trivial stabilizer code with a transversal non-Clifford gate.This compact code underlies the ququint protocol’s practical advantages.
  • Ququint protocol: 36.3% depolarizing-noise threshold is achieved by the ququint protocol.The threshold is reported as ϵ*dep = 0.363, and the protocol has better thresholds than other protocols with polynomially scaling yield.
  • Performance: The ququint protocol has higher yield than all known qubit protocols, according to numerical and analytic scaling comparisons.For dimensions d > 5, thresholds and resource costs deteriorate with increasing dimension, although the source leaves the cause unresolved.
  • Performance: The 8-qutrit protocol is less effective than the ququint protocol but remains competitive with qubit protocols.The paper examines this qutrit scheme in detail alongside the ququint protocol.
  • Comparisons and scope: Comparing noise thresholds across dimensions is unsettled because both ϵ and the depolarizing rate δ have fairness concerns.The authors note that δ penalizes higher-dimensional states, while higher-dimensional systems also have more noise processes contributing to depolarization.
  • Comparisons and scope: The results motivate further study of complete qutrit- and ququint-based fault-tolerance schemes, whose comparable thresholds are currently unknown.Enhanced magic-state-distillation performance may or may not translate into better full-scheme thresholds and resource costs.
  • Applications: The protocols may also apply when fault-tolerant operations form a proper subgroup of the Clifford group, including qudit topological-cluster models.The authors specifically anticipate applications to XZ eigenstate preparation and state that the protocols can distill XZ stabilizer states.

Appendix A: The canonical M gate

Appendix A verifies that the canonical gate M satisfies the conditions required for membership in the relevant M_m gate sets. The argument establishes integrality for the asserted prime dimensions and treats the d = 3, m = 1 case separately.

  • Canonical gate construction: The canonical M gate is shown to satisfy the requirements for membership in M_m.The appendix explicitly states this conclusion after verifying the defining conditions.
  • Integrality conditions: For m ≥ 2, the construction gives integer parameters for all prime dimensions d.The integrality checks for c and related quantities are obtained by inspection or the preceding arithmetic argument.
  • Integrality conditions: For m = 1, the construction is integer-valued for prime d ≥ 5 because factors such as (d−1)(d−2) are divisible by the required denominators.The appendix uses divisibility by 6 and 24 to establish the needed integer values.
  • Exceptional case: For d = 3 and m = 1, the construction does not establish a member of M_1, and numerical search finds no non-Clifford gate in that set.The text attributes the failure first to non-integral λ_j values and then reports the numerical search result.
  • Distillation analysis: The Reed–Muller-code analysis distinguishes detected errors, no-error instances, and undetected errors through projection onto logical states.Detected errors cause the projected state to vanish, while no-error instances project onto the same logical state; other cases yield other logical states.
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