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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
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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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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.