In 2024, Kliuchnikov and Schönnenbeck showed a connection between the Barnes Wall lattices, stabilizer states and Clifford operations. In this work, we study their results and relate them to the problem of lower bounding stabilizer ranks. We show the first quantitative lower bound on stabilizer fidelity as a function of stabilizer ranks, which reproduces the linear-by-log lower bound for χ δ ( | H ⟩ ⊗ n ) , i.e, on the approximate stabilizer rank of | H ⟩ ⊗ n . In fact, we show that the lower bound holds even when the fidelity between the approximation and | H ⟩ ⊗ n is exponentially small, which is currently the best lower bound in this regime.Next, we define a new magic monotone for pure states, the Barnes Wall norm, and its corresponding approximate variant. We upper bound these monotones by the C S -count of state preparation, and also by the stabilizer ranks. In particular, the upper bound given by the C S -count is tight, in the sense that we exhibit states that achieve the bound.Apart from these results, we give a Fidelity Amplification algorithm, which provides a trade-off between approximation error and the stabilizer rank. As a corollary, it gives us a way to compose approximate stabilizer decompositions into approximate decompositions of their tensor products.Finally, we provide an alternate, elementary proof of the existence and density of product states with maximal stabilizer ranks, which was first proven by Lovitz and Steffan (2022), where they used results from algebraic geometry.In 2024, Kliuchnikov and Schönnenbeck showed a connection between the Barnes Wall lattices, stabilizer states and Clifford operations. In this work, we study their results and relate them to the problem of lower bounding stabilizer ranks. We show the first quantitative lower bound on stabilizer fidelity as a function of stabilizer ranks, which reproduces the linear-by-log lower bound for χ δ ( | H ⟩ ⊗ n ) , i.e, on the approximate stabilizer rank of | H ⟩ ⊗ n . In fact, we show that the lower bound holds even when the fidelity between the approximation and | H ⟩ ⊗ n is exponentially small, which is currently the best lower bound in this regime.Next, we define a new magic monotone for pure states, the Barnes Wall norm, and its corresponding approximate variant. We upper bound these monotones by the C S -count of state preparation, and also by the stabilizer ranks. In particular, the upper bound given by the C S -count is tight, in the sense that we exhibit states that achieve the bound.Apart from these results, we give a Fidelity Amplification algorithm, which provides a trade-off between approximation error and the stabilizer rank. As a corollary, it gives us a way to compose approximate stabilizer decompositions into approximate decompositions of their tensor products.Finally, we provide an alternate, elementary proof of the existence and density of product states with maximal stabilizer ranks, which was first proven by Lovitz and Steffan (2022), where they used results from algebraic geometry.

arXiv:2503.04101QuantumJul 29, 20260 citations

Quantum lower bounds by entropy numbers

Stefan Heinrich

arXiv:quant-ph/0611294Journal of ComplexityMar 14, 20079 citations

Distributed quantum approximate counting algorithm

Huaijing Huang, Daowen Qiu

Jan 1, 20260 citations

Quantitative and Optimal Device-Independent Lower Bounds on Detection Efficiency

Arkaprabha Ghosal, Soumyadip Patra, Peter Bierhorst

This paper examines a quantitative and optimal lower bound on the detector efficiency in a (2,2,2) Bell experiment within a fully device-independent framework, whereby the detectors used in the experiment are uncharacterized. We provide a tight lower bound on the minimum efficiency required to observe a desired Bell-CHSH violation using the Navascués-Pironio-Acín (NPA) hierarchy, confirming tightness up to four decimal places with numerical optimization over explicit quantum realizations. We then introduce the effect of dark counts and demonstrate how to quantify the minimum required efficiency to observe a desired CHSH violation with an increasing dark count error. Finally, to obtain an analytical closed-form expression of the minimum efficiency, we consider the set of no-signaling behaviors that satisfy the Tsirelson bound, which are easier to characterize than the quantum set. Using such behaviors, we find a simple closed-form expression for a lower bound on the minimum efficiency which is monotonically increasing with the CHSH violation, though the analytically obtained lower bounds are meaningfully below the numerically tight lower bound.This paper examines a quantitative and optimal lower bound on the detector efficiency in a (2,2,2) Bell experiment within a fully device-independent framework, whereby the detectors used in the experiment are uncharacterized. We provide a tight lower bound on the minimum efficiency required to observe a desired Bell-CHSH violation using the Navascués-Pironio-Acín (NPA) hierarchy, confirming tightness up to four decimal places with numerical optimization over explicit quantum realizations. We then introduce the effect of dark counts and demonstrate how to quantify the minimum required efficiency to observe a desired CHSH violation with an increasing dark count error. Finally, to obtain an analytical closed-form expression of the minimum efficiency, we consider the set of no-signaling behaviors that satisfy the Tsirelson bound, which are easier to characterize than the quantum set. Using such behaviors, we find a simple closed-form expression for a lower bound on the minimum efficiency which is monotonically increasing with the CHSH violation, though the analytically obtained lower bounds are meaningfully below the numerically tight lower bound.

quant-pharXiv:2511.19302Physical Review ANov 25, 20250 citations

New Bounds for Quick Computation of the Lower Bound on the Gate Count of Toffoli-Based Reversible Logic Circuits

Takashi HIRAYAMA, Rin Suzuki, Katsuhisa Yamanaka +1 more

We present a time-efficient lower bound κ on the number of gates in Toffoli-based reversible circuits that represent a given reversible logic function. For the characteristic vector s of a reversible logic function, κ(s) closely approximates σ-lb(s), which is known as a relatively efficient lower bound in respect of evaluation time and tightness. The primary contribution of this paper is that κ enables fast computation while maintaining a tightness of the lower bound, approximately equal to σ-lb. We prove that the discrepancy between κ(s) and σ-lb(s) is at most one only, by providing upper and lower bounds on σ-lb in terms of κ. Subsequently, we show that κ can be calculated more efficiently than σ-lb. An algorithm for κ(s) with a complexity of O(n) is presented, where n is the dimension of s. Experimental results comparing κ and σ-lb are also given. The results demonstrate that the two lower bounds are equal for most reversible functions, and that the calculation of κ is significantly faster than σ-lb by several orders of magnitude.We present a time-efficient lower bound κ on the number of gates in Toffoli-based reversible circuits that represent a given reversible logic function. For the characteristic vector s of a reversible logic function, κ(s) closely approximates σ-lb(s), which is known as a relatively efficient lower bound in respect of evaluation time and tightness. The primary contribution of this paper is that κ enables fast computation while maintaining a tightness of the lower bound, approximately equal to σ-lb. We prove that the discrepancy between κ(s) and σ-lb(s) is at most one only, by providing upper and lower bounds on σ-lb in terms of κ. Subsequently, we show that κ can be calculated more efficiently than σ-lb. An algorithm for κ(s) with a complexity of O(n) is presented, where n is the dimension of s. Experimental results comparing κ and σ-lb are also given. The results demonstrate that the two lower bounds are equal for most reversible functions, and that the calculation of κ is significantly faster than σ-lb by several orders of magnitude.

IEICE Transactions on Information and SystemsJul 31, 20241 citation

Robustness of Quantum Signal Processing and Resource-aware Polynomial Design

Thiago O. Maciel, Giancarlo Camilo, Allan Tosta +3 more

Quantum signal processing (QSP) realizes operator functions through polynomial transformations of a finite degree. In addition to function approximation errors, realistic implementations incur algorithmic errors also when encoding the input operator and in synthesizing gates into a discrete fault-tolerant gate set. We provide an end-to-end analysis of these errors and show that the input error propagation is governed by the operator-Lipschitz constant $L_\star$ of the implemented function. We then derive an efficient classical algorithm to compute tight lower and upper bounds on $L_\star$ , and we show that these bounds induce in turn lower and upper bounds on the total quantum-resource cost (for instance, $T$ -gate count) needed for QSP implementations compatible with a target total error. We further devise a procedure to search for a polynomial with minimum cost based solely on its coefficients, yielding a resource-aware polynomial-design procedure for QSP. We illustrate the resulting method for ground state energy estimation via eigenvalue thresholding for the H $_6$ molecular Hamiltonian. In the tested hardware profiles, the resource-aware family improves the total-error budget and reduces the modeled $T$ -gate count by $51.9\%$ -- $56.7\%$ relative to the standard analytical polynomial obtained from an error-function approximation.Quantum signal processing (QSP) realizes operator functions through polynomial transformations of a finite degree. In addition to function approximation errors, realistic implementations incur algorithmic errors also when encoding the input operator and in synthesizing gates into a discrete fault-tolerant gate set. We provide an end-to-end analysis of these errors and show that the input error propagation is governed by the operator-Lipschitz constant $L_\star$ of the implemented function. We then derive an efficient classical algorithm to compute tight lower and upper bounds on $L_\star$, and we show that these bounds induce in turn lower and upper bounds on the total quantum-resource cost (for instance, $T$-gate count) needed for QSP implementations compatible with a target total error. We further devise a procedure to search for a polynomial with minimum cost based solely on its coefficients, yielding a resource-aware polynomial-design procedure for QSP. We illustrate the resulting method for ground state energy estimation via eigenvalue thresholding for the H$_6$ molecular Hamiltonian. In the tested hardware profiles, the resource-aware family improves the total-error budget and reduces the modeled $T$-gate count by $51.9\%$--$56.7\%$ relative to the standard analytical polynomial obtained from an error-function approximation.

quant-pharXiv:2610.03885Oct 6, 20260 citations

L_\star

of the implemented function. We then derive an efficient classical algorithm to compute tight lower and upper bounds on

L_\star

, and we show that these bounds induce in turn lower and upper bounds on the total quantum-resource cost (for instance,

T

-gate count) needed for QSP implementations compatible with a target total error. We further devise a procedure to search for a polynomial with minimum cost based solely on its coefficients, yielding a resource-aware polynomial-design procedure for QSP. We illustrate the resulting method for ground state energy estimation via eigenvalue thresholding for the H

_6

molecular Hamiltonian. In the tested hardware profiles, the resource-aware family improves the total-error budget and reduces the modeled

T

-gate count by

51.9\%

56.7\%

relative to the standard analytical polynomial obtained from an error-function approximation.