Introduces antiflatness of entanglement spectra, antiflat majorization based on Rényi entropy spread, and unifies measures via escort distributions while connecting capacity of entanglement to quantum Fisher information.
Non-Clifford Cost of Random Unitaries
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abstract
Recent years have enjoyed a strong interest in exploring properties and applications of random quantum circuits. In this work, we explore the ensemble of $t$-doped Clifford circuits on $n$ qubits, consisting of Clifford circuits interspersed with $t$ single-qubit non-Clifford gates. We establish rigorous convergence bounds towards unitary $k$-designs, revealing the intrinsic cost in terms of non-Clifford resources in various flavors. First, we analyze the $k$-th order frame potential, which quantifies how well the ensemble of doped Clifford circuits is spread within the unitary group. We prove that a quadratic doping level, $t = \tilde{\Theta}(k^2)$, is both necessary and sufficient to approximate the frame potential of the full unitary group. As a consequence, we refine existing upper bounds on the convergence of the ensemble towards state $k$-designs. Second, we derive tight bounds on the convergence of $t$-doped Clifford circuits towards relative-error $k$-designs, showing that $t = \tilde{\Theta}(nk)$ is both necessary and sufficient for the ensemble to form a relative $\varepsilon$-approximate $k$-design. Similarly, $t = \tilde{\Theta}(n)$ is required to generate pseudo-random unitaries. All these results highlight that generating random unitaries is extremely costly in terms of non-Clifford resources, and that such ensembles fundamentally lie beyond the classical simulability barrier. Additionally, we introduce doped-Clifford Weingarten functions to derive analytic expressions for the twirling operator over the ensemble of random doped Clifford circuits, and we establish their asymptotic behavior in relevant regimes.
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Exact results show U(1) symmetry substantially suppresses non-stabilizerness in random states, with different leading scaling from entanglement near zero charge density.
A new framework certifies global quantum properties including multipartite entanglement, circuit complexity, and quantum magic on small subsystems with constant sample complexity via local Pauli measurements.
Demonstrates a task solvable with 12 qubits but requiring 62-382 classical bits of memory, yielding unconditional quantum information supremacy on a trapped-ion processor.
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.
citing papers explorer
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A journey through Flatland: What does the antiflatness of a spectrum teach us?
Introduces antiflatness of entanglement spectra, antiflat majorization based on Rényi entropy spread, and unifies measures via escort distributions while connecting capacity of entanglement to quantum Fisher information.
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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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Certifying localizable quantum properties with constant sample complexity
A new framework certifies global quantum properties including multipartite entanglement, circuit complexity, and quantum magic on small subsystems with constant sample complexity via local Pauli measurements.
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Demonstrating an unconditional separation between quantum and classical information resources
Demonstrates a task solvable with 12 qubits but requiring 62-382 classical bits of memory, yielding unconditional quantum information supremacy on a trapped-ion processor.
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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.