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Quantifying the magic of quantum channels

Xin Wang, Mark M. Wilde, Yuan Su

arXiv:1903.04483v2quant-phcs.ITmath-ph

TL;DR

Universal fault-tolerant computation needs non-stabilizer resources beyond stabilizer operations, motivating a resource theory for magic quantum channels. The paper uses CPWP operations as free operations and introduces efficiently computable channel measures, showing limits on magic distillation, a four-T lower bound for CCX synthesis, and mana-based noisy-circuit simulation.

  • Problem

    Stabilizer operations are efficiently classically simulable and do not enable universal computation, creating a need to quantify non-stabilizerness in noisy quantum operations.

  • Method

    The paper develops an odd-prime-dimensional qudit resource theory with CPWP free operations, mana and max-thauma channel measures, and a mana-based simulation algorithm.

  • Results

    The measures bound amortized and distillable magic, imply that at least four T gates are required for perfect CCX implementation, and parameterize a simulation algorithm that can outperform channel robustness.

  • Takeaways & Limitations

    The framework establishes limits and operational tools for processing quantum magic in noisy circuits and fault-tolerant quantum computation.

Abstract

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To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum "magic" or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension $d$, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.

I. INTRODUCTION

Universal fault-tolerant quantum computation requires non-stabilizer resources beyond classically simulable stabilizer operations. This paper develops a resource theory for magic quantum channels in odd-prime-dimensional qudit systems, introducing CPWP operations and channel measures with applications to distillation, synthesis, and simulation.

  • Motivation: Stabilizer operations are insufficient for universal computation because they can be efficiently simulated classically, motivating quantitative measures of non-stabilizerness.Non-stabilizer operations can supply the additional resource needed for universality.
  • Framework: The framework studies quantum channels on qudit systems with odd prime dimension and takes completely positive-Wigner-preserving operations as the free operations.This free-operation set is larger than the one used in related work.
  • Channel measures: The paper introduces the mana and max-thauma of quantum channels and proves faithfulness, tensor-product additivity, serial-composition subadditivity, amortization inequalities, and CPWP-superchannel monotonicity.These measures generalize magic quantification from states to channels.
  • Applications: Both amortized magic and distillable magic of a channel are bounded above by its mana and max-thauma.These bounds quantify limits on the ability of channels to generate magic states.
  • Applications: At least four T gates are required to perfectly implement a controlled-controlled-NOT gate.The channel measures are applied to magic cost and quantum gate synthesis.
  • Applications: A classical simulation algorithm for noisy quantum circuits has sample complexity scaling with channel mana and can outperform a channel-robustness approach in concrete examples.The algorithm targets noisy circuits relevant to NISQ devices.

C. Stabilizer channels and beyond

The section broadens stabilizer-based resource theories to completely positive-Wigner-preserving operations, characterized through channel Wigner representations and reference-system preservation.

  • Stabilizer operations include Clifford operations, stabilizer-state ancillas, partial trace, computational-basis measurements, and classically conditioned post-processing.
  • Completely PWP operations are motivated as free operations because circuits with non-negative Wigner functions can be classically simulated efficiently.
  • The conditional Wigner representation propagates input Wigner functions to output Wigner functions, including when a reference system is present.
  • CPWP, non-negative Choi-Jamiołkowski Wigner function, and non-negative conditional channel Wigner function are equivalent characterizations.
  • The CPWP characterization extends earlier equivalence results by incorporating information processing in the presence of reference systems.

B. Quantum (CPWP) superchannels

CPWP superchannels extend completely Wigner-preserving operations to transformations of quantum channels, using bipartite-channel and non-signaling representations.

  • A quantum superchannel physically transforms an input channel into an output channel and corresponds one-to-one with a bipartite channel obeying a non-signaling constraint.
  • The bipartite representation imposes complete positivity, trace preservation, and prevention of signaling from system B to system A.
  • The discrete Wigner function of a superchannel is a conditional quasi-probability distribution with an additional non-signaling constraint.
  • CPWP superchannels are defined to map every CPWP channel, including channels with arbitrary reference systems, to another CPWP channel.
  • CPWP superchannels are equivalently characterized by a non-negative Choi Wigner function or a non-negative, non-signaling conditional Wigner distribution.
  • Every CPWP superchannel has a non-unique realization through pre-processing and post-processing CPWP channels.

C. Logarithmic negativity (mana) of a quantum channel

The channel mana is a logarithmic-negativity measure of non-stabilizerness with structural properties including faithfulness, composition bounds, and monotonicity under CPWP superchannels.

  • The mana of a quantum channel is introduced as the channel’s logarithmic negativity based on its discrete Wigner representation.
  • Mana is additive under tensor products and subadditive under serial composition: M(N1 ⊗ N2) = M(N1) + M(N2), while M(N2◦N1) ≤ M(N1) + M(N2).
  • For replacer channels, mana reduces to the mana of the output state.
  • Mana is faithful: M(N) ≥ 0, and M(N) = 0 if and only if N is CPWP.
  • The amortization inequality bounds the magic generated by applying a channel to a state using the channel mana plus the input-state mana.
  • Mana is monotone under CPWP superchannels, and therefore also under completely stabilizer-preserving superchannels.

D. Generalized thauma of a quantum channel

The generalized thauma extends state-based magic measures to quantum channels by minimizing a generalized divergence over completely positive maps with non-positive mana. It is shown to reduce correctly to states, be monotone and faithful, and admit a minimax formulation under suitable divergence conditions.

  • Definition: The generalized thauma minimizes a generalized channel divergence over completely positive maps E satisfying M(E) ≤ 0.The construction extends the generalized thauma previously defined for quantum states.
  • Reduction to states: For replacer channels, the generalized thauma reduces to the corresponding generalized thauma of the prepared output state.A replacer channel acts as N(ρ) = Tr[ρ]σ for an arbitrary input state.
  • Resource properties: The generalized thauma is monotone under completely positive-Wigner-preserving superchannels.The proof uses data processing for generalized channel divergences and monotonicity of mana under the superchannel.
  • Resource properties: For strongly faithful divergences, the generalized thauma is non-negative and vanishes on CPWP channels; with continuity, vanishing occurs only for CPWP channels.Thus, under the stated continuity condition, the measure is faithful.
  • Minimax formulation: Continuous, direct-sum-preserving, second-argument-convex divergences permit exchanging the minimization over free maps and maximization over pure channel inputs.The result follows by concavity, convexity, and the Sion minimax theorem.

E. Max-thauma of a quantum channel

The max-thauma specializes generalized thauma to the max-relative entropy and has an SDP formulation. It satisfies reduction, monotonicity, faithfulness, tensor-product additivity, serial-composition subadditivity, and amortization bounds.

  • Definition and computation: The max-thauma is the max-relative entropy divergence between a channel and completely positive maps with non-positive mana.It can be expressed as a semidefinite program using the channel’s Choi–Jamiołkowski matrix.
  • Resource properties: Max-thauma is non-negative and equals zero if and only if the channel is CPWP.It is also monotone under CPWP superchannels and reduces to the state measure for replacer channels.
  • Comparison with mana: The max-thauma does not exceed the mana of the channel.The comparison follows directly from the primal SDP formulation.
  • Composition properties: θmax(N1 ⊗ N2) = θmax(N1) + θmax(N2) for tensor products of quantum channels.The proof combines primal and dual SDP constructions with additivity of max-relative entropy.
  • Composition properties: Max-thauma is subadditive under serial composition and obeys amortization inequalities for channel outputs and reference-system inputs.These properties imply that arbitrary pre- and post-processing does not increase max-thauma.

A. Amortized magic

Amortized magic measures how much magic a channel can generate, while distillable magic measures rates for extracting target magic states through repeated channel uses and free interleaving operations. The paper bounds these capabilities using channel magic measures, including an upper bound on T-state distillation rates.

  • Amortized magic: Amortized magic is the largest increase in a magic measure produced when a channel acts on an arbitrary input state.Strict amortized magic restricts the input to a stabilizer state.
  • Bounds: The mana and max-thauma provide upper bounds on the amortized magic of a quantum channel.The bounds follow from amortization inequalities for the corresponding state measures.
  • Distillation protocol: The general distillation protocol uses repeated channel invocations interleaved with CPWP channels, producing target magic states at rate k/n.The protocol starts from a state with non-negative Wigner function and ends with high-fidelity copies of the target state.
  • T gate: The qutrit T gate generates its associated T magic state from |+⟩, while state injection implements the gate using stabilizer operations and the T state.This connects gate distillation and magic-state distillation for the channel considered.
  • T-state distillation: The max-thauma also upper-bounds the rate R = k/n of an (n, k, ε) T-magic distillation protocol and the channel’s T-distillable magic.The proof uses max-relative entropy, CPWP monotonicity, and the channel amortization inequality.

C. Injectable quantum channel

Injectable channels can be implemented using a CPWP operation and an associated resource state, reducing channel-resource questions to state-resource questions. For resource-seizable channels, the resulting bounds become equalities; the T channel is a central example.

  • Definition: An injectable channel is realized by applying a CPWP channel to the input together with an associated resource state.The defining relation holds for every input state.
  • Resource-state reduction: For injectable channels, channel mana and generalized thauma are bounded by the corresponding measures of the associated resource state.The bounds use circuit injection, data processing, and multiplicativity of the Wigner trace norm.
  • Resource seizure: If an injectable channel is also resource seizable, the channel-measure bounds become equalities.Resource seizure uses free pre-processing and CPWP post-processing to recover the associated resource state.
  • T-channel example: For the T channel, θmax(T) = θ(T) = θmax(|T⟩⟨T|) = θ(|T⟩⟨T|) = log(1 + 2 sin(π/18)).The T channel is both injectable and resource seizable with associated resource state |T⟩⟨T|.
  • Distillation bounds: For injectable channels, distillation through repeated channel uses reduces to state distillation from repeated copies of the associated resource state.This reduction yields improved upper bounds on T-magic distillation rates.

V. MAGIC COST OF A QUANTUM CHANNEL

This section develops channel-synthesis costs under CPWP assistance, deriving magic-based lower bounds for exact and approximate simulation. It applies these bounds to qutrit CCX synthesis and noisy quantum-circuit simulation.

  • Exact channel simulation: Exact synthesis asks how many uses of a channel N′, supplemented by CPWP channels, are required to implement a target channel N.If exact simulation is impossible, the synthesis cost is defined as infinite.
  • Exact channel simulation: Magic measures provide lower bounds on the number of resource channels required for exact channel synthesis, including bounds based on mana and related quantities.The bounds follow from applying the channel measures to arbitrary synthesis protocols with CPWP operations.
  • Gate synthesis: At least four qutrit T gates are required to synthesize a controlled-controlled-X qutrit gate exactly.The result is established as a direct application of the channel-magic bounds and numerical evaluation.
  • Noisy gate synthesis: Under depolarizing noise, the framework yields a lower bound on the number of noisy T gates needed to implement a low-noise CCX gate.For depolarizing noise with p = 0.01, the bound is expressed using the magic of the noisy T channel and the target CCX channel.
  • Approximate channel simulation: Approximate channel simulation is treated by minimizing exact-simulation magic costs over channels within diamond-norm error ε of the target.The resulting lower bound can be computed through a semidefinite program.
  • Noisy-circuit simulation: Mana quantifies the classical simulation cost of noisy circuits, with sample complexity controlled by the mana of the circuit’s quantum channels.The proposed sampling procedure produces an unbiased estimate and requires enough samples to achieve specified accuracy ε and success probability 1 −δ.

B. Comparison of classical simulation algorithms for noisy quantum circuits

This section compares mana-based and channel-robustness approaches for simulating noisy qudit circuits. The mana-based method is never worse under the stated odd-prime-dimension setting and can be strictly faster for some circuits.

  • Simulation setting: The comparison considers noisy circuits on n qudits of odd prime dimension, with channels decomposed relative to completely stabilizer-preserving operations.Both simulation approaches target computational-basis measurement probabilities with prescribed accuracy and success probability.
  • Measure comparison: Exponentiated mana is always no greater than channel robustness, and the inequality can be strict.The paper establishes the inequality using the channel-robustness decomposition and gives an example demonstrating strict separation.
  • Measure comparison: For the diagonal-unitary example, mana is strictly smaller than channel robustness and magic capacity over the plotted parameter range.Figure 5 compares M_Uθ with R_W+(Φ_Uθ) for π ≤θ ≤2π; the gap indicates strict separation from C(Uθ) and R∗(Uθ).
  • Simulation performance: For odd-prime-dimensional qudit circuits, the mana-based algorithm’s sample complexity is never worse than the channel-robustness approach.The demonstrated separation further shows that mana can make the simulation strictly faster for certain circuits.
  • Scope: The general relation between mana and magic capacity remains unclear beyond the example establishing a strict separation.The paper explicitly leaves open whether the observed ordering holds for general quantum channels.

VII. EXAMPLES

The examples examine how depolarizing noise suppresses non-stabilizerness in T and CCX gates, and characterize the qutrit Werner–Holevo channel's magic behavior. They also connect channel measures to classical simulation cost and broader limits on processing quantum magic.

  • A. Non-stabilizerness under depolarizing noise: At p ≥ 0.62, the depolarized T channel D_p ◦ T cannot generate non-stabilizerness and becomes CPWP.
  • A. Non-stabilizerness under depolarizing noise: The mana of D_p^⊗3 ◦ CCX decreases linearly with p and reaches zero at approximately p ≈ 0.75.
  • B. Werner-Holevo channel: The Werner–Holevo channel maps every quantum state to a free state with non-negative Wigner function, while its amortized magic equals the channel measure.
  • A. Non-stabilizerness under depolarizing noise: Figure 7's solid line also quantifies the classical simulation cost of the noisy CCX circuit D_p^⊗3 ◦ CCX.
  • VIII. CONCLUSION: The paper presents channel measures as tools for evaluating magic generation, gate synthesis, and classical simulation, while establishing limitations on processing quantum magic.
  • VIII. CONCLUSION: Tighter evaluations of distillable channel magic and extensions to multiqubit systems remain future directions.

Appendix A: On completely positive maps with non-positive mana

The appendix establishes technical properties of completely positive maps with non-positive mana and uses them in the resource-theoretic characterization of free channels.

  • If a completely positive map E has M(E) ≤ 0, then E is trace non-increasing on states with non-negative Wigner function.
  • Lemma 30 states an inequality for any operator Q and CPWP channel Π.
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