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Phase transition in magic with random quantum circuits

Pradeep Niroula, Christopher David White, Qingfeng Wang, Sonika Johri, Daiwei Zhu, Christopher Monroe, Crystal Noel, Michael J. Gullans

arXiv:2304.10481v2quant-ph

TL;DR

The paper investigates how coherent errors and syndrome measurements shape magic in random stabilizer codes, a resource for universal fault-tolerant computation. It analyzes the resulting phases and finds a transition between non-magical and magical behavior, with syndrome measurements suppressing magic below threshold and concentrating it above. The authors also discuss how this behavior may inform magic-state generation, while noting that their measurements address small systems.

  • Problem

    Understanding how magic is created or destroyed matters because magic is a resource for universal fault-tolerant quantum computation.

    Source: §V · p. 7

  • Method

    The study probes random Clifford codes subject to coherent errors with analytic and numerical calculations and experimental measurements.

    Source: §IV · p. 4, §V · p. 7

  • Results

    αc=0.27(1), with scaling collapse indicating a phase transition from non-magical to magical behavior; syndrome measurements suppress magic below threshold and concentrate it above.

    Source: §IV · p. 4, §V · p. 7

  • Takeaways & Limitations

    In the magical phase, syndrome measurements move magic from physical qubits to logical qubits, which may provide states for existing magic-state distillation protocols if the required decoding is solved.

    Source: §V · p. 7

    The study measures magic in small systems because current measures are difficult to evaluate at large sizes, and larger-system diagnostics remain a future possibility.

    Source: §V · p. 7

Abstract

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Magic is a property of quantum states that enables universal fault-tolerant quantum computing using simple sets of gate operations. Understanding the mechanisms by which magic is created or destroyed is, therefore, a crucial step towards efficient and practical fault-tolerant computation. We observe that a random stabilizer code subject to coherent errors exhibits a phase transition in magic, which we characterize through analytic, numeric and experimental probes. Below a critical error rate, stabilizer syndrome measurements remove the accumulated magic in the circuit, effectively protecting against coherent errors; above the critical error rate syndrome measurements concentrate magic. A better understanding of such rich behavior in the resource theory of magic could shed more light on origins of quantum speedup and pave pathways for more efficient magic state generation.

I. MODEL

The model encodes an all-zero product state with a random Clifford circuit, applies coherent single-qubit rotations, then decodes and measures the syndrome qubits. The study compares vanishing-rate and constant-rate codes, using different circuit depths in numerics and experiments.

II. MODEL

  • The circuit encodes |0⟩⊗N with a random Clifford C, applies Rz(α) to every qubit, decodes with C†, and measures N−K syndrome qubits, leaving K logical qubits.The rotation angle α is the error rate.Source: §I · p. 2
  • The encoder interweaves d layers of single-qubit Clifford gates with d layers of entangling gates on randomly chosen disjoint pairs, and the decoder is its inverse.Numerics use d=N, while experiments use d=N/2 to reduce noise effects; the shallower circuits behave similarly to depth N.Source: §I · p. 2
  • Coherent rotations move the encoded stabilizer state off the stabilizer grid, while decoding and syndrome measurement can return it to the grid or leave a magical logical state.The final state is in the logical space in either case.Source: §I · p. 2
  • For vanishing-rate codes K=1 and r=1/N tends to zero as N grows; for constant-rate codes K=rN at fixed r.Here r is the ratio of logical to physical qubits.Source: §I · p. 2

II. QUANTIFYING MAGIC

The paper quantifies magic using the second stabilizer Rényi entropy and basis-minimized measurement entropy. The latter minimizes measurement-outcome entropy across stabilizer bases, while conditional entropy tracks logical uncertainty given the measured syndrome.

II. QUANTIFYING MAGIC

  • The second stabilizer Rényi entropy measures how spread out a state's density matrix is over Pauli operators: it is zero for stabilizer states and nearly maximal for Haar states.For N-qubit Haar states it is proportional to N.Source: §II · p. 2
  • Basis-minimized measurement entropy minimizes the Born-outcome entropy over stabilizer measurement bases and is zero for stabilizer states.It is also non-increasing under Clifford unitaries and subadditive for product states.Source: §II · p. 2
  • The basis-minimized conditional entropy averages logical measurement entropy over syndrome outcomes and minimizes it over logical stabilizer bases.The conditional entropy without basis minimization restricts measurement to the computational basis.Source: §II · pp. 2–3
  • The conditional entropy without basis minimization is expected to approximate its minimized value at small α, when the logical state remains close to its initial computational-basis state.At α=0 and α=π/2, both experimental magic measures are zero because the final states are stabilizer states.Source: §II · p. 3
  • Conditional entropy measures logical-space uncertainty given a syndrome and directly bounds a decoder's ability to recover encoded classical information from logical-qubit measurements.The basis-minimized form allows all Clifford decoding operations; the unminimized form limits the decoder to computational-basis measurements.Source: §II · p. 3

III. MAGIC IN THE VANISHING RATE CODE

For a single logical qubit, magic is finite between the two Clifford points and peaks near α=π/2, with the peak approaching that point as system size grows. The study derives a square-root scaling near the Clifford point and reports experimental measurements that qualitatively resemble theoretical expectations.

III. MAGIC IN THE VANISHING RATE CODE

  • For one logical qubit, SSRE is finite between α=0 and π/2 and peaks a distance proportional to 1/N from α=π/2.At large N, the Clifford point becomes a singularity.Source: §III · pp. 3–4
  • Near α=π/2, unequal error weights for the two errors producing a syndrome create amplitude imbalance and magic, with syndrome-averaged SSRE scaling as N(π/2−α)^2.The error weights are drawn from a binomial distribution; simulations and experiments show the predicted square-root scaling.Source: §III · p. 4
  • The experiments group syndromes by their effective logical action because direct postselection on syndrome outcomes is prohibitively expensive.Tomography is performed on the single logical qubit, and density matrices are projected to their maximum-eigenvalue eigenstates to mitigate incoherent errors.Source: §III · p. 4
  • The measured magic qualitatively resembles theoretical expectations for N=8, 12, and 16 after mitigation, using 50, 50, and 30 random-circuit instances, respectively.The experimental data are from a 16-qubit processor; error bars use bootstrap resampling.Source: §III · p. 4

IV. CONSTANT RATE

At constant code rate, the finite-magic critical region of vanishing-rate codes becomes an extended magical phase. Scaling collapses indicate a phase transition, while entropy diagnostics yield distinct critical estimates that the authors do not unify.

IV. CONSTANT RATE

  • The scaling collapse indicates that the transition from non-magical to magical is a phase transition, not a crossover.Source: §IV · p. 4
  • Conditional entropy is used as a phase diagnostic because finite-rate SSRE is costly to measure, but without basis minimization it only upper-bounds a genuine magic measure.At small α the computational basis is expected to minimize entropy; at the Clifford point, another stabilizer basis gives zero measurement entropy while the computational-basis entropy is extensive.Source: §IV · p. 4
  • For the Rényi conditional measurement entropy, analytics match exact numerics for large systems when the typicality assumption holds.The paper also reports that this Rényi diagnostic exhibits phases similar to the Shannon conditional measurement entropy.Source: §IV · p. 5
  • At r=1/2, the SSRE collapse gives αc/π=0.27(1), ν=1.15(4), and γ=1.20(8) as the vanishing-rate critical region extends into a magical phase.The authors note that the exponent estimates depend somewhat on the regime used for the collapse.Source: §IV · p. 4, Figure 3 · p. 6
  • For r=1/2, conditional entropy gives αc/π=0.304(2), ν=2.9(2), while its Rényi approximation gives αc/π=0.347(1), ν=2.6(2); whether their transitions coincide remains unresolved.The authors say the estimates are difficult to establish confidently and depend strongly on the scaling hypothesis.Source: §IV · p. 5, Figure 3 · p. 6
  • IonQ Aria measurements show entropy scaling inverted across the threshold, with scaling collapses using critical exponents from simulations of d=N/2 circuits.The experiment used circuits up to 16 qubits; conditional measurement entropy was accessible through computational-basis measurements, unlike finite-rate SSRE.Source: §IV · pp. 6–7

V. DISCUSSION

Random Clifford codes with coherent errors exhibit a magic phase transition: syndrome measurements suppress magic below a critical value and concentrate it above. The discussion connects this behavior to decoder breakdown and possible magic-state generation, while noting limits in system size and unresolved universality.

V. DISCUSSION

  • Syndrome measurements suppress magic below a critical error or code rate and concentrate it above the critical value in random Clifford codes with coherent errors.This connects the magic transition to decoder breakdown through basis-minimized measurement entropy.Source: §V · p. 7
  • The study covers small systems, where calculating magic measures is tractable; such measures are generally difficult to estimate for large systems.The authors suggest conditional entropy as a possible efficient diagnostic for larger systems.Source: §V · p. 7
  • The authors describe the transition as competition among noncommuting channels that generate the resource, generate correlations, and destroy the resource.For this transition, encoding Clifford operations generate correlations, Rz(α) rotations generate magic, and syndrome measurements destroy it.Source: §V · p. 7
  • It remains unclear whether the studied magic phase transitions belong to a universality class of critical phenomena.Source: §V · p. 7
  • The authors suggest coherent noise and error correction may create useful magic states, but identifying the appropriate syndrome-dependent Clifford recovery remains a challenge.If recovery can be solved more generally, the magical phase may improve magic-state distillation efficiency.Source: §V · p. 7

DATA AVAILABILITY

The paper identifies a Zenodo repository for its data and includes a disclaimer about identifying commercial equipment, instruments, or materials.

DATA AVAILABILITY

DISCLAIMER

  • Identifying commercial equipment, instruments, or materials specifies the experimental procedure and does not imply endorsement or that those items are best available.Source: DISCLAIMER · p. 7

9 Supplementary Material

The supplementary material characterizes basis-minimized measurement entropy as a measure of magic, relates it to decoder breakdown, and analyzes syndrome distributions and the effective logical action of coherent errors. Its vanishing-rate analysis estimates magic near and away from the Clifford point.

A. BASIS-MINIMIZED MEASUREMENT ENTROPY AS A MEASURE OF MAGIC

  • Basis-minimized measurement entropy minimizes the Shannon entropy of measurement outcomes over stabilizer bases and is zero for stabilizer states, non-increasing under Clifford operations, and subadditive.The supplementary argument establishes faithfulness, Clifford stability, and subadditivity.Source: A. BASIS-MINIMIZED MEASUREMENT ENTROPY… · p. 9

B. DISTRIBUTION OF SYNDROMES IN VANISHING RATE AND FINITE RATE CODES

C. BASIS-MINIMIZED MEASUREMENT ENTROPY AND DECODER BREAKDOWN

  • Basis-minimized conditional entropy probes optimal Clifford-decoder breakdown: large measurement entropy indicates that classical information may be irrecoverably muddled, while small entropy does not guarantee recovery.Syndrome measurement can project the logical state to a pure stabilizer state different from the initial one.Source: C. BASIS-MINIMIZED MEASUREMENT ENTROPY… · pp. 10–11

D. ANALYTICAL ESTIMATE OF MAGIC IN THE VANISHING RATE CODE

  • For the vanishing-rate code, the expected magic follows a function of Nϵ², with the supplied analysis specifying the small-Nϵ² regime.The section estimates magic for a single logical qubit.Source: D. ANALYTICAL ESTIMATE OF MAGIC IN THE… · p. 11

1. The Clifford point

  • At the Clifford point, the effective logical action is unitary in specified error-pair cases and otherwise a projector; the unitary case is expected with probability 1/2.The analysis postselects codes with exactly two errors per syndrome.Source: §1 · p. 12

2. Away from the Clifford point

  • Away from the Clifford point, the effective action remains unitary for the specified error-pair cases and is otherwise unitarily equivalent to a weak projection, with the unitary cases occurring at approximately probability 1/2.Source: §2 · p. 12

B. Magic

The section explains how errors and syndrome measurements affect magic, then describes numerical scaling analyses and circuit-depth choices used to study the transition. It also gives finite-size scaling-collapse procedures for estimating critical parameters.

B. Magic

  • When the effective operation K acts as a weak projector, an initial logical stabilizer state remains a stabilizer state; anticommuting Pauli terms instead move it away from one.Near α = π/2, the displacement scales as (nb − na)ϵ, and the second Rényi entropy of magic is evaluated from this rotation.Source: B. Magic · pp. 12–13

E. NUMERICS FOR d = N/2

  • Experiments use d = N/2 circuits to reduce noise effects; their critical exponents differ from d = N circuits, while d = 2N yields exponents close to d = N.The authors expect exponents to converge for sufficiently deep encoders and suggest d = N captures ensemble-average properties.Source: E. NUMERICS FOR d = N/2 · p. 13
  • For d = N/2, the conditional-entropy collapse gives αc/π = 0.300(2), ν = 1.40(6), while its Rényi approximation gives αc/π = 0.351(1), ν = 1.24(4).For d = 2N, the corresponding values are αc/π = 0.302(3), ν = 2.9(2), and αc/π = 0.348(3), ν = 2.8(4).Source: Figure S2 · p. 14, Figure S3 · p. 14

G. FINITE SIZE SCALING

  • The scaling analysis assumes f(α, N) expands in x = (α − αc)N^(1/ν), then fits αc and ν by minimizing squared residuals against numerical data.Errors are estimated by omitting each datapoint in turn; the error in ν is obtained similarly.Source: G. FINITE SIZE SCALING · pp. 14–15

H. BOOTSTRAP ESTIMATE OF ERROR

The section describes bootstrap error estimates and an analytic route to the circuit-averaged Rényi analogue of conditional entropy. The analytic treatment uses Clifford averaging and a typicality assumption that can fail for small systems at large error rates.

H. BOOTSTRAP ESTIMATE OF ERROR

  • Bootstrap error bars are estimated from 1,000 replacement-resampled samples of size 20, using the standard deviation of their means.The original sample has size N.Source: H. BOOTSTRAP ESTIMATE OF ERROR · p. 15

I. ANALYTICS ON R ´ENYI-APPROXIMATION TO CONDITIONAL ENTROPY

  • Moving the circuit expectation inside the logarithm assumes negligible collision-probability variability, which holds for sufficiently large systems but breaks down for small N at large α approaching π/2.The calculation proceeds using this typicality assumption.Source: §I · p. 15
  • The analytic calculation decomposes a Clifford-averaged operator into representations of the 30-element semigroup Σ4 and evaluates coefficients through simultaneous equations.The resulting contributions are combined to obtain the expected collision probability and approximate the circuit-averaged Rényi conditional entropy.Source: §I · pp. 16–17

J. r →1 LIMIT

The r → 1 limit is analytically tractable because syndrome measurements are absent, and its magic density remains constant as N grows. The section also specifies the noise model used for vanishing-rate noisy simulations.

J. r →1 LIMIT

  • As N → ∞, the r → 1 magic density remains constant, implying volume-law scaling regardless of system size.The r → 1 limit has no syndrome measurements, and the analytical result is presented alongside numerical SSRE for tractable sizes.Source: J. r →1 LIMIT · pp. 17–18, Figure S4 · p. 18

K. NOISY SIMULATION

  • Noisy simulations apply independent Gaussian XX over-rotations with ϵ sampled from N(0, 0.06), followed by depolarization at p = 0.016, and compare with L = 8, 12 experiments.The noise model is applied after every two-qubit XX gate.Source: K. NOISY SIMULATION · p. 18

L. DATA WITHOUT POST-PROCESSING

The section presents constant-rate experimental conditional entropies under varying post-processing. It describes projection onto the ideal subspace and a comparison involving measured and theoretically calculated probabilities.

L. DATA WITHOUT POST-PROCESSING

  • Constant-rate experimental data undergoes post-processing that projects the measured probability distribution onto the ideal noiseless wavefunction's distribution.Conditional Shannon and Rényi entropies are presented with varying degrees of post-processing.Source: L. DATA WITHOUT POST-PROCESSING · p. 18, Figure S6 · p. 19
  • The section contrasts unmitigated data with procedures that project onto the ideal subspace and use either measured or theoretically calculated probabilities.The supplied passages identify these processing variants but do not state their comparative outcomes.Source: Figure S6 · p. 19, L. DATA WITHOUT POST-PROCESSING · p. 19
  • One post-processing measure computes −Σx∈Ideal p(x) log ˜p(x), using measured probabilities p(x) and theoretically calculated probabilities ˜p(x).The measured distributions are projected onto the support of the ideal distribution.Source: L. DATA WITHOUT POST-PROCESSING · p. 19

M. CONSTRAINED SCALING COLLAPSE OF CONSTANT-RATE MAGIC PHASE TRANSITION

The appendix constrains the constant-rate magic scaling collapse because its unconstrained γ≈1.2 fit is deemed unphysical for the available system sizes. The constrained r=1/2 collapse uses γ=1.000.

M. CONSTRAINED SCALING COLLAPSE OF CONSTANT-RATE MAGIC PHASE TRANSITION

  • The free scaling collapse yielded γ≈1.2, which the authors regard as unphysical and attribute to the small system sizes available.They therefore constrain γ to the expected range [0,1].Source: M. CONSTRAINED SCALING COLLAPSE OF… · p. 19
  • γ=1.000 is used for the constrained r=1/2 SSRE scaling collapse, within the expected physical range [0,1].The inset scales the error-rate axis by (α/π − αc)N^(1/ν), with αc=0.240(2) and ν=0.96(5).Source: M. CONSTRAINED SCALING COLLAPSE OF… · p. 19, Figure S8 · p. 20
  • In Fig. S8, error bars indicate standard error and are omitted from the scaling-collapse inset.Source: Figure S8 · p. 20
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