StabilizerBench is a new benchmark for evaluating AI agents on generating, optimizing, and making fault-tolerant stabilizer circuits for quantum error correction, with efficient verification and multi-tier scoring.
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The Heisenberg Representation of Quantum Computers
Canonical reference. 85% of citing Pith papers cite this work as background.
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abstract
Since Shor's discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key features of quantum computers - the difficulty of describing them on classical computers - also makes it difficult to describe and understand precisely what can be done with them. A formalism describing the evolution of operators rather than states has proven extremely fruitful in understanding an important class of quantum operations. States used in error correction and certain communication protocols can be described by their stabilizer, a group of tensor products of Pauli matrices. Even this simple group structure is sufficient to allow a rich range of quantum effects, although it falls short of the full power of quantum computation.
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- abstract Since Shor's discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key features of quantum computers - the difficulty of describing them on classical computers - also makes it difficult to describe and understand precisely what can be done with them. A formalism describing the evolution of operators rather than states has proven extremely fruitful in understanding an important class of quantum operations. States used in error correction and certain communication protocols can
- method Section 5 concludes with a summary of results and future directions. 2 Frame-Factored State Representation Standard state vector simulation scales exponentially with the total number of qubits in the system. However, fault-tolerant quantum circuits are typically dominated by Clifford operations, which can be tracked efficiently via the Gottesman-Knill theorem [37]. We utilize this structure by shifting from the standard Schrödinger picture to a hybrid representation that decouples the state into
- background [40] Z. Wang,Topological Quantum Computation, American Mathematical Society (2010). [41] M.A. Nielsen and I.L. Chuang,Quantum Computation and Quantum Information, Cambridge University Press, 10th anniversary edition ed. (2010). [42] S. Aaronson and D. Gottesman,Improved simulation of stabilizer circuits,Phys. Rev. A70(2004) 052328 [quant-ph/0406196v5]. [43] D. Gottesman,The heisenberg representation of quantum computers, in22nd International Colloquium on Group Theoretical Methods in Physics, pp
- background as starting points to include correlations, while the modern tools of tensor networks offer controlled, low-entanglement expansions around tensor-product states. Another notable class of states in the Hilbert space, that was only recently introduced in the many body-physics area [30], is that of stabilizer states. As first pointed out by Gottesmann and Knill [31], these states are special in that they can be prepared efficiently with classical computing [31,32], while being able to support high
- background The study of the complexity of classical algorithms informs us that some problems cannot be solved by computers in reasonable timeframes. Quantum computers can solve these problems quickly, and the construction of these devices will bring profound technological changes in the upcoming decades. Some practical applications for which quantum algorithms currently exist include molecular simulation for drug discovery and material design [BGM+19, LBG+21, SBW+21, RBK+23], algorithms that break public-k
- background advantage using photons, Science370, 1460 (2020), https://www.science.org/doi/pdf/10.1126/science.abe8770. [4] D. Gottesman, The heisenberg representation of quantum computers, arXiv preprint quant-ph/9807006 (1998). [5] S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal clifford gates and noisy ancillas, Physical Review A71, 022316 (2005). [6] A. Mari and J. Eisert, Positive wigner functions ren- der classical simulation of quantum computation ef- ficient, Physical Review Lette
- background this by applying probabilistic inequalities such as Cheby- shev's inequality, Pr ∣ψ⟩∼ELU(ψ0) [∣sE(ψ)−∥E∥2 F∣≥kσ] ≤ 1 k2 , whereσ= √ Var(sE(ψ))denotes the standard de- viation. Error Kurtosis and Quantum Magic-In magic- state resource theory, Clifford operations and stabilizer states are considered "free" as a direct consequence of the Gottesman-Knill theorem [13, 14], which establishes their efficient classical simulability. In contrast, non- Clifford gates-though essential for achieving univers
co-cited works
representative citing papers
Single-qudit universality for Clifford gate sets plus one non-Clifford gate follows a trichotomy determined by the prime factorization of the local dimension d.
The Triangle Criterion detects mixed-state magic, proves multi-qubit distillation is strictly stronger than single-qubit schemes, and identifies a purity bound plus undetectable unfaithful magic states.
Sdim is the first open-source qudit stabilizer simulator supporting all dimensions, enabling circuit evaluation and sampling for qudit fault-tolerant quantum computing research.
Non-Hermitian quantum circuits with renormalization after fixed non-unitary gates are equivalent to PostBQP, which equals PP, in the uniform circuit model.
CRiSP uses neural-guided MCTS and curriculum learning to insert Clifford prefixes before parameterized rotations in VQAs, yielding mean 3.17x and max 45x gains in energy accuracy on 22-qubit QAOA benchmarks versus prior Clifford initializers.
Stabilizer Rényi entropy provides an exact closed-form witness for CP phases in spin-0 decays that standard entanglement quantifiers miss, with linear and quartic magic-inspired observables proposed for collider use.
Above a critical noise strength, operator scrambling in random circuits is suppressed leading to classical simulability; below it, simulation stays exponentially hard.
A randomized linear-time phase-folding algorithm using constant-width bitstring abstraction optimizes T-count in quantum circuits orders of magnitude faster than prior tools while achieving comparable reductions.
In U(1)-symmetric random circuits, initial states with lower stabilizer Rényi entropy generate nonstabilizerness faster than those with higher entropy, with the effect also depending on spatial charge structure and extending to SU(2) circuits and Hamiltonian dynamics.
A Monte Carlo Tree Search with GNN-based magic estimation biases quantum circuit search toward target nonstabilizerness levels and yields better results on ground-state energy and state approximation problems.
Clifft introduces a factored-state simulator that shifts exponential cost to a dynamic active subspace, generalizing Stim's compile-once model to near-Clifford circuits and enabling the first exact end-to-end simulations of magic-state cultivation over hundreds of billions of shots.
Nonlocal magic in fermionic Gaussian states is bounded by the entanglement spectrum of the covariance matrix, is extensive in the Haar ensemble, peaks at criticality in the Kitaev chain, and grows diffusively under random circuits.
Reset-induced entanglement phase transitions in measurement-free random quantum circuits are continuous for d=2 with second-order characteristics, unlike large-d classical expectations.
Decoherence of the color code produces a mixed state with topological entanglement negativity ln 2 that corresponds to an emergent single toric code.
A discrete phase-space path integral is constructed for finite quantum mechanics, reducing to classical deterministic flow for linear Hamiltonians while requiring all fluctuation sectors to capture entanglement dynamics in qutrit systems.
Current-state opacity is formalized in safe partially observed quantum Petri nets with true-concurrency semantics and verified exactly via stabilizer formalism and targeted unfolding.
Exact results show U(1) symmetry substantially suppresses non-stabilizerness in random states, with different leading scaling from entanglement near zero charge density.
Circle graphs are closed under r-local complementation and bipartite circle graph states correspond one-to-one with planar code states whose MBQC is classically simulable.
Stabilizer Rényi entropy of 3-uniform hypergraph states equals a matrix-rank expression, cutting computation from exponential in 3N to polynomial in N times exponential in N.
A systematic analysis of 59 quantum software testing empirical studies reveals highly diverse designs, inconsistent reporting, and open methodological challenges, leading to recommendations for future work.
A sampling method combining fast Walsh-Hadamard transform and Clifford-preconditioned Monte Carlo reduces Pauli-string sampling cost from O(2^N) to O(N) with sample count independent of N for stabilizer Rényi entropies and nullity.
The stabilizer Rényi entropy governs the exponential rate at which Clifford orbits become indistinguishable from Haar-random states and sets the optimal distinguishability from stabilizer states in property testing.
Analytical and numerical study of stabilizer nullity and Rényi entropies in monitored Clifford circuits shows quantized decay for computational measurements and size-dependent relaxation to a non-trivial steady state for rotated bases.
citing papers explorer
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StabilizerBench: A Benchmark for AI-Assisted Quantum Error Correction Circuit Synthesis
StabilizerBench is a new benchmark for evaluating AI agents on generating, optimizing, and making fault-tolerant stabilizer circuits for quantum error correction, with efficient verification and multi-tier scoring.
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Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources
Single-qudit universality for Clifford gate sets plus one non-Clifford gate follows a trichotomy determined by the prime factorization of the local dimension d.
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Triangle Criterion: a mixed-state magic criterion with applications in distillation and detection
The Triangle Criterion detects mixed-state magic, proves multi-qubit distillation is strictly stronger than single-qubit schemes, and identifies a purity bound plus undetectable unfaithful magic states.
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Sdim: A Qudit Stabilizer Simulator
Sdim is the first open-source qudit stabilizer simulator supporting all dimensions, enabling circuit evaluation and sampling for qudit fault-tolerant quantum computing research.
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Computational Complexity and Simulability of Non-Hermitian Quantum Dynamics
Non-Hermitian quantum circuits with renormalization after fixed non-unitary gates are equivalent to PostBQP, which equals PP, in the uniform circuit model.
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Classical State Preparation for Variational Quantum Algorithms via Reinforcement Learning
CRiSP uses neural-guided MCTS and curriculum learning to insert Clifford prefixes before parameterized rotations in VQAs, yielding mean 3.17x and max 45x gains in energy accuracy on 22-qubit QAOA benchmarks versus prior Clifford initializers.
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Quantum Magic Reveals CP Phases Invisible to Entanglement in Spin-0 Decays
Stabilizer Rényi entropy provides an exact closed-form witness for CP phases in spin-0 decays that standard entanglement quantifiers miss, with linear and quartic magic-inspired observables proposed for collider use.
-
Noise-induced Simulability Transition from Operator Scrambling
Above a critical noise strength, operator scrambling in random circuits is suppressed leading to classical simulability; below it, simulation stays exponentially hard.
-
Linear-Time T-Gate Optimization via Random Abstraction
A randomized linear-time phase-folding algorithm using constant-width bitstring abstraction optimizes T-count in quantum circuits orders of magnitude faster than prior tools while achieving comparable reductions.
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Nonstabilizerness Mpemba Effects
In U(1)-symmetric random circuits, initial states with lower stabilizer Rényi entropy generate nonstabilizerness faster than those with higher entropy, with the effect also depending on spatial charge structure and extending to SU(2) circuits and Hamiltonian dynamics.
-
Magic-Informed Quantum Architecture Search
A Monte Carlo Tree Search with GNN-based magic estimation biases quantum circuit search toward target nonstabilizerness levels and yields better results on ground-state energy and state approximation problems.
-
Clifft: Fast Exact Simulation of Near-Clifford Quantum Circuits
Clifft introduces a factored-state simulator that shifts exponential cost to a dynamic active subspace, generalizing Stim's compile-once model to near-Clifford circuits and enabling the first exact end-to-end simulations of magic-state cultivation over hundreds of billions of shots.
-
Nonlocal nonstabilizerness in free fermion models
Nonlocal magic in fermionic Gaussian states is bounded by the entanglement spectrum of the covariance matrix, is extensive in the Haar ensemble, peaks at criticality in the Kitaev chain, and grows diffusively under random circuits.
-
Continuous Reset-Induced Phase Transition in Measurement-Free Random Quantum Circuits
Reset-induced entanglement phase transitions in measurement-free random quantum circuits are continuous for d=2 with second-order characteristics, unlike large-d classical expectations.
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Decohered color code and emerging mixed toric code by anyon proliferation: Topological entanglement negativity perspective
Decoherence of the color code produces a mixed state with topological entanglement negativity ln 2 that corresponds to an emergent single toric code.
-
Path integral formulation of finite-dimensional quantum mechanics in discrete phase space
A discrete phase-space path integral is constructed for finite quantum mechanics, reducing to classical deterministic flow for linear Hamiltonians while requiring all fluctuation sectors to capture entanglement dynamics in qutrit systems.
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Current-State Opacity in Safe Partially Observed Quantum Petri Nets: True-Concurrency Semantics and Exact Symbolic Verification
Current-state opacity is formalized in safe partially observed quantum Petri nets with true-concurrency semantics and verified exactly via stabilizer formalism and targeted unfolding.
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Non-stabilizerness and U(1) symmetry in chaotic many-body quantum systems
Exact results show U(1) symmetry substantially suppresses non-stabilizerness in random states, with different leading scaling from entanglement near zero charge density.
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The Structure of Circle Graph States
Circle graphs are closed under r-local complementation and bipartite circle graph states correspond one-to-one with planar code states whose MBQC is classically simulable.
-
Stabilizer R\'enyi entropy of 3-uniform hypergraph states
Stabilizer Rényi entropy of 3-uniform hypergraph states equals a matrix-rank expression, cutting computation from exponential in 3N to polynomial in N times exponential in N.
-
A Methodological Analysis of Empirical Studies in Quantum Software Testing
A systematic analysis of 59 quantum software testing empirical studies reveals highly diverse designs, inconsistent reporting, and open methodological challenges, leading to recommendations for future work.
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Exponentially Accelerated Sampling of Pauli Strings for Nonstabilizerness
A sampling method combining fast Walsh-Hadamard transform and Clifford-preconditioned Monte Carlo reduces Pauli-string sampling cost from O(2^N) to O(N) with sample count independent of N for stabilizer Rényi entropies and nullity.
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Operational interpretation of the Stabilizer Entropy
The stabilizer Rényi entropy governs the exponential rate at which Clifford orbits become indistinguishable from Haar-random states and sets the optimal distinguishability from stabilizer states in property testing.
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Rise and fall of nonstabilizerness via random measurements
Analytical and numerical study of stabilizer nullity and Rényi entropies in monitored Clifford circuits shows quantized decay for computational measurements and size-dependent relaxation to a non-trivial steady state for rotated bases.
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Disentangling strategies and entanglement transitions in unitary circuit games with matchgates
Introduces a minimal matchgate circuit representation for fermionic Gaussian states together with a Yang-Baxter update algorithm, then maps out entanglement transitions in unitary circuit games under braiding and generic matchgate rules.
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Bra-ket entanglement, an indicator bridging entanglement, magic, and coherence
Bra-ket entanglement indicates a shift from coherence-dominated to magic-dominated entanglement generation as its value increases.
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Classical simulability of Clifford+T circuits with Clifford-augmented matrix product states
Develops an optimization-free disentangling algorithm and algebraic criterion for efficient CAMPS representations of Clifford circuits doped with αI+βP gates, enabling polynomial classical simulation for more circuits including typical N-T-gate random instances.
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Magic state cultivation: growing T states as cheap as CNOT gates
Magic state cultivation prepares high-fidelity T states with an order of magnitude fewer qubit-rounds than prior distillation methods by gradually growing them within a surface code under depolarizing noise.
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Quantum resource redistribution drives spectral splits in dense neutrino gases
Tensor network simulations of two-flavor neutrinos link spectral splits to peaks in entanglement entropy and local minima in non-local magic, indicating resource redistribution drives the phenomenon.
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The relative entropy of magic and its nonadditivity
Characterizes qubit magic states via relative entropy of entanglement results and proves nonadditivity of relative entropy of magic for multi-qubit tensor products.
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Detrimental Agnostic Entanglement: The Case Against Hardware-Efficient Ans\"atze for Combinatorial Optimization
For diagonal Hamiltonians like MaxCut, hardware-efficient ansatze drive entanglement down during training and are outperformed by separable circuits in a monotonic relationship, while QAOA's problem-derived entanglement remains competitive.
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Quantum magic of strongly correlated fermions $-$ the Hubbard dimer
Non-stabilizerness of the Hubbard dimer is computed with robustness of magic and stabilizer Rényi entropy, revealing it as a resource distinct from fermionic non-Gaussianity and superselected entanglement.
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The true cost of factoring: Linking magic and number-theoretic complexity in Shor's algorithm
Shor's algorithm generates and consumes magic resources in direct proportion to the difficulty of the underlying factoring problem.
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Quantum Magic in early FTQC: From Diagonal Clifford Hierarchy No-Go Theorems to Architecture Design Blueprints
No-go theorems prove hierarchy level and state-independent sequences cannot maximize operational magic in early FTQC, requiring state-aware differentiable optimization and nonlinear phases for scalable magic generation.
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Interplay of Nonstabilizerness and Ergotropy in Quantum Batteries
Ergotropy in the battery corresponds one-to-one with total nonstabilizerness under U(1)-symmetric charger-battery interactions, while maximum average charging power in Clifford evolution is achievable even with zero initial magic.
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Non-Local Magic Resources for Fermionic Gaussian States
Closed-form formula computes non-local magic for fermionic Gaussian states from two-point correlations in polynomial time.
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Continuous Noise Model for Quantum Circuits
Continuous coherent noise modeled via von Mises-Fisher rotations degrades logical performance in quantum error-correcting codes more than equivalent Pauli noise.
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Equivalence Checking of Quantum Circuits via Path-Sum and Weighted Model Counting
A hybrid path-sum reduction plus weighted model counting method yields a complete equivalence checker for quantum circuits up to global phase.
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Complexity of quantum states in the stabilizer formalism
A complexity quantifier for stabilizer quantum states is defined via Jordan and Lie products and linked to nonstabilizerness via the L^4-norm of characteristic functions.
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Orkan: Cache-friendly simulation of quantum operations on hermitian operators
Orkan simulates quantum operations on Hermitian operators using a cache-friendly tiled lower-triangle layout, halving memory and achieving 2-4x speedups over Qiskit Aer, QuEST, and Qulacs.
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Quantifying magic via quantum $(\alpha,\beta)$ Jensen-Shannon divergence
Two new magic quantifiers are introduced using quantum (α,β) Jensen-Shannon divergence, shown to have desirable properties and efficient computation in low dimensions, with initial magic boosting generation under certain gates.
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PAEMS: Precise and Adaptive Error Model for Superconducting Quantum Processors
PAEMS is a new adaptive qubit error model that reduces timelike, spacelike, and spacetime error correlations by 19.5×, 9.3×, and 5.2× on IBM QPUs while outperforming Google's SI1000 model by 58-73% across multiple platforms.
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Universal Non-stabilizerness Dynamics Across Quantum Phase Transitions
Stabilizer Rényi entropies and Pauli spectrum cumulants show universal power-law scaling with driving rate in slow processes across quantum phase transitions, with the logarithmic Pauli spectrum asymptotically Gaussian, demonstrated in the transverse-field Ising model and long-range Kitaev models.
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Emulation of large-scale qubit registers with a phase-space approach
A mean-field phase-space method emulates continuous-time dynamics of up to thousands of qubits with quadratic cost, capturing single-qubit observables qualitatively on transverse-field Ising models.
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Sample- and Hardware-Efficient Fidelity Estimation by Stripping Phase-Dominated Magic
Phase stripping reduces target-state magic to enable O(poly(n)) or O(1) sample fidelity estimation for phase-dominated states using a single fan-out gate plus nonlinear Pauli post-processing.
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Stabilizer Code-Generic Universal Fault-Tolerant Quantum Computation
Ancilla-mediated protocols enable deterministic universal logical gates on any stabilizer code without ancilla consumption or code modification.
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Exact and Efficient Stabilizer Simulation of Thermal-Relaxation Noise for Quantum Error Correction
An exact positive-probability decomposition of thermal relaxation noise into Clifford gates and resets exists for T2 ≤ T1, with a negativity-free approximation that outperforms Pauli twirling for T2 > T1.
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A Framework for Quantum Simulations of Energy-Loss and Hadronization in Non-Abelian Gauge Theories: SU(2) Lattice Gauge Theory in 1+1D
A quantum simulation framework is developed and demonstrated for energy loss and hadronization of a heavy quark in 1+1D SU(2) lattice gauge theory on 18 qubits of IBM hardware, with results matching classical simulations.
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Multiqubit Rydberg Gates for Quantum Error Correction
Global multiqubit Rydberg gates enable break-even measurement-free QEC and lower-shuttling Floquet codes in neutral-atom hardware.
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Automorphism in Gauge Theories: Higher Symmetries and Transversal Non-Clifford Logical Gates
Automorphisms of gauge groups extend to higher or non-invertible symmetries in topological gauge theories and enable transversal non-Clifford gates in 2+1d Z_N qudit Clifford stabilizer models for N greater than or equal to 3.