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Measuring magic on a quantum processor
Salvatore F. E. Oliviero, Lorenzo Leone, Alioscia Hamma, Seth Lloyd
TL;DR
Quantum advantage requires magic, but hardware needs a way to measure this resource and diagnose deviations caused by imperfections. The paper introduces and experimentally demonstrates randomized local measurements of stabilizer 2-Rényi entropy, using circuits with injected non-Clifford gates on IBM processors. The measurements characterize hardware-produced magic and support noise modeling, while the protocol’s faithful estimation has an exponentially demanding accuracy requirement in qubit number.
Problem
Quantum advantage requires resources beyond classically simulable Clifford circuits, motivating a method to quantify magic and characterize whether hardware produces it accurately.
Method
The paper uses randomized local Clifford measurements to estimate stabilizer 2-Rényi entropy and purity in circuits containing injected non-Clifford gates.
Results
The experiments show that measuring stabilizer Rényi entropy characterizes the magic produced by quantum circuits and that inaccurate magic levels signal hardware errors.
Takeaways & Limitations
Magic measurement can contribute to quantum-hardware certification by identifying unwanted magic associated with imperfect gate implementation.
Takeaways & Limitations
Faithful measurement of the stabilizer purity requires error ϵ ∼ d−1, making the required resources exponentially large in the number of qubits.
Abstract
from arXiv · showhide
Magic states are the resource that allows quantum computers to attain an advantage over classical computers. This resource consists in the deviation from a property called stabilizerness which in turn implies that stabilizer circuits can be efficiently simulated on a classical computer. Without magic, no quantum computer can do anything that a classical computer cannot do. Given the importance of magic for quantum computation, it would be useful to have a method for measuring the amount of magic in a quantum state. In this work, we propose and experimentally demonstrate a protocol for measuring magic based on randomized measurements. Our experiments are carried out on two IBM Quantum Falcon processors. This protocol can provide a characterization of the effectiveness of a quantum hardware in producing states that cannot be effectively simulated on a classical computer. We show how from these measurements one can construct realistic noise models affecting the hardware.
INTRODUCTION
The paper addresses how to characterize quantum hardware and measure the magic resource needed beyond efficiently classically simulable Clifford computation. It proposes an experimentally demonstrated randomized-measurement protocol for stabilizer 2-Rényi entropy, avoiding tomography and reducing measurement complexity.
- Magic is the resource beyond Clifford operations required for quantum computational advantage, while Clifford circuits can be efficiently simulated classically.
- Accurately measuring magic matters because decoherence and gate imperfections can increase or decrease it, with excess unwanted magic signaling noise.
- The paper proposes and experimentally demonstrates randomized measurements to measure magic and characterize quantum hardware.
- The protocol measures stabilizer 2-Rényi entropy, whose direct evaluation would otherwise require all multi-qubit Pauli expectation values and tomography-scale resources.
The protocol
The protocol prepares states with controlled non-Clifford content, applies randomized local Clifford operations, and estimates magic-related quantities from computational-basis measurements. It uses local randomized measurements rather than global unitary sampling and evaluates circuits with injected T-gate resources on IBM processors.
- Randomized measurement correlations estimate the stabilizer quantity and purity, while Clifford 2-design properties allow both quantities to be obtained from occupation probabilities.
- The protocol uses local Clifford-group randomization instead of full-unitary randomization, reducing reliance on noisier multi-qubit operations.
- Experiments were conducted on two IBM Quantum Falcon processors: the 5-qubit ibmq_quito and 7-qubit ibmq_casablanca.
- The experiment prepares an n-qubit state, applies independently sampled one-qubit Clifford operations, and measures in the computational basis.
- The test circuits are t-doped Clifford circuits, where injected Pϑ gates provide magic and the T gate at ϑ = π/4 is the maximal-magic choice for this gate family.
Measuring magic
The experiments measure stabilizer 2-Rényi entropy across single-qubit and entangled states, varying the number of injected T-gates on IBM processors. For larger systems, decoherence and gate imperfections cause measured magic to depart from ideal predictions, enabling noise characterization.
- Single-qubit states: Single-qubit measurements on ibmq_quito agree with theoretical magic values, while purity remains unity within experimental errors.These results indicate negligible decoherence for n = 1.
- Entangled states: The prepared entangled states |Γ^(n)_t⟩ vary the number of T-gates across doped random Clifford circuits for n = 3, 4, and 5.The circuits use up to 2n − 1 magic seeds, with resource counts parameterized separately for each system size.
- Entangled states: For larger n, state purity is compromised by decoherence and measured magic increasingly departs from the theoretical pure-state prediction, especially at low magic.The measured value can exceed the ideal value because imperfect Clifford gates or decoherence may inject or subtract unwanted magic.
- Noise characterization: The experiments use stabilizer 2-Rényi entropy to characterize how accurately processors create the intended amount of magic, rather than merely whether they create magic.This accuracy requirement allows the measured deviations to estimate noise-model parameters.
- Noise characterization: The noise model combines state-preparation decoherence with unitary imperfections in randomized-measurement gates and is tuned to match experimental deviations.Its predicted magic provides a better approximation to the data as the number of T-gates increases.
DISCUSSION
The paper concludes that quantum hardware must produce an accurate amount of magic, because implementation errors can inject or subtract unwanted magic and degrade computation. Measuring stabilizer Rényi entropy can therefore help certify hardware and identify unitary errors.
- Imperfect Clifford gates can inject or subtract uncontrolled magic, so creating magic alone is insufficient for reliable computation.
- Magic measurements provide an additional tool for evaluating experimental setups and certifying quantum hardware.
- Uncontrolled excess magic can degrade a circuit’s ability to perform a desired task, analogous to excess entanglement.
- Comparing measured magic with its theoretical value can signal unitary errors beyond decoherence.
Theoretical framework
The theoretical framework defines stabilizer 2-Rényi entropy through purity and stabilizer purity, then estimates both with randomized local Clifford measurements. This replaces global multi-qubit measurements with single-qubit operations and computational-basis readout.
- The protocol measures stabilizer 2-Rényi entropy from the purity P(ψ) and stabilizer purity W(ψ).
- Local randomized measurements reconstruct the operators needed for P(ψ) and W(ψ) using single-qubit gates and computational-basis measurements.
- For each sampled local Clifford C, repeated computational-basis measurements estimate occupation probabilities from the resulting state.
- The estimated probabilities are combined with weighting coefficients to compute P(ψ) and W(ψ).
- The local construction uses single-qubit diagonal operators whose Clifford averages reproduce the swap and four-copy operators.
Statistical analysis
The statistical analysis separates errors from finite local-Clifford sampling and finite measurement shots, then selects resource allocations by simulation. Resource requirements depend on precision, system size, and the number of injected T-gates.
- Finite local-Clifford sampling and finite measurement shots are the two sources of statistical error.
- In the large-dimension, large-shot regime, the estimator variance scales inversely with the number of sampled unitaries.
- The total resources for stabilizer purity scale as O(ϵ−2) for error ϵ under the stated statistical bound.
- Faithful stabilizer-purity estimation requires ϵ ∼ d−1, making the resource count exponentially large in the number of qubits.
- The experiment uses NU random local Clifford unitaries and NM projective measurements per unitary to estimate purity and stabilizer purity.
- The fitted curves are reported to agree perfectly with the experimental data within the grid’s finite resolution.
- The optimal resource pair minimizes NUNM subject to distance and purity thresholds, and depends on the target state.
- For fixed n, simulations fit the optimal total resources to NTOT = 2a+b[(2n−1)−t], showing dependence on injected T-gates.
Supplementary Information for “Measuring magic on a quantum processor”
The paper is authored by researchers affiliated with the University of Massachusetts Boston, the University of Naples Federico II and INFN, MIT, and Turing Inc.
- The authors are affiliated with five institutions across the United States and Italy.
I. SUPPLEMENTARY NOTE 1: THE STABILIZER R´ENYI ENTROPY: A WRAP-UP OF THE RESULTS
The stabilizer Rényi entropy quantifies magic and connects it to quantum certification, information extraction, and chaos. Its 2-Rényi version is experimentally accessible through randomized measurements and relates magic to distinguishability and measurement resources.
- The stabilizer Rényi entropy quantifies the resources needed to extract useful information from a quantum state.
- The α-Stabilizer Rényi entropy is defined from a Pauli-operator probability distribution combined with the state purity.
- The 2-Stabilizer Rényi entropy is experimentally measurable through statistical correlations between randomized measurements.
- More magic in a unitary's Choi state corresponds to more chaotic evolution, and α = 2 enables direct measurement of an 8-point OTOC.
- Extensive magic can make distinguishing a state from a random state exponentially difficult, while low Pauli distinguishability requires many measurement shots.
- Stabilizer Rényi entropies directly quantify resources required for Monte Carlo fidelity estimation.
A. Magic is robust under noisy preparations
The paper analyzes how noise affects magic in high-magical, entangled states using a model based on Pauli errors. Under the stated noise conditions, the model predicts controlled magic changes and motivates experimentally fitting the noise distribution.
- The noise analysis models a high-magical, entangled target as a Haar-random state and studies the average magic change under noisy preparation.
- The derivation uses typicality of purity-related quantities and incurs an error exponentially small in n.
- 1 ≤ X ≤ d^2, with X = d^2 for unitary stabilizer noise and X = 1 for a completely dephasing channel.
- For large d, the noisy state's averaged stabilizer purity scales as αd^-2 with 1 ≤ α ≤ 4.
- The experimental data indicate that the noise distribution has 2-Rényi entropy S2(q) ≤ O(poly(log n)), and the model sets S2(q) = O(log n).
B. Noise model
The noise model separates state-preparation decoherence from measurement-gate imperfections and estimates their parameters from randomized-measurement data. Purity measurements provide a state-aware decoherence parameter, while calibration on an unprepared computational-basis state determines readout and gate-error parameters.
- Each experiment prepares a state, applies random local Clifford gates, and performs local projective measurements.
- State-preparation decoherence is modeled by phase flips on qubits, with a state-dependent parameter p governing the error probability.
- Random Clifford-gate imperfections are modeled by a small phase displacement ε applied to each local gate.
- The measured stabilizer purity combines the noisy state's purity with a measurement-apparatus error term depending on ε.
- Purity estimation is protected against gate-imperfection errors and can therefore determine the state-aware preparation-noise parameter p.
- The parameter p is obtained from the positive solution of a second-order equation, and p = 1 when experimental purity equals one.
- Readout error is calibrated on the computational-basis state, where q = 1 if the measured purity is one.