arXiv subjects

Lenny Liu

Publications and source records attributed to Lenny Liu.

5 recordsLinked to original sources

arXiv2026

PureSuperQMA(exp) = BellPureSymQMA(poly) = QMA via Dimension-Free Bosonic Argmax

Pure-state consistency problems naturally lead to quantum proof systems in which a single pure witness must satisfy many acceptance constraints. The corresponding class $\mathsf{PureSuperQMA}$ was previously known to lie between $\mathsf{QMA}$ and $\mathsf{QMA}(2)$, and Kamminga and Rudolph (ITCS'26) conjectured that both containments are strict. In this paper, we prove the following surprising complexity collapses $$ \mathsf{QMA} = \mathsf{PureSuperQMA} = \mathsf{PureSuperQMA}(\text{exp}) = \mathsf{BellPureSymQMA}(\text{poly}) $$ Here $\mathsf{PureSuperQMA}(\text{exp})$ allows exponentially many checks which are uniformly indexed and efficiently generated, while requiring an inverse-polynomial violation margin and an inverse-polynomial fraction of violated checks for the NO cases. $\mathsf{BellPureSymQMA}(\text{poly})$ is a related model that requires the prover to give the verifier polynomially many copies of a pure state, which the verifier measures separately with logarithmic output length for each local measurement, before processing the outcomes jointly. The main technical ingredient is a dimension-free stability bound for symmetric tensor states. Our simulations use polynomially many witness registers and combine a random-pair SWAP test with a permutation-invariant lift of the original verification procedure. The key step is to show that, on the symmetric subspace, the extremal verification value is close to that of some tensor-power witness with dimension-independent error. Applying this argument to the two verification models yields both simulations. As a consequence, exact $k$-local pure-state consistency is $\mathsf{QMA}$-complete for every fixed $k\ge2$, and so are the corresponding exact bosonic and fermionic pure $N$-representability problems.

quant-ph↗

arXiv2026

The Honeycomb Framework for Code Bounds

We introduce the honeycomb hierarchy, a representation-theoretic framework that gives new asymptotic upper bounds on $R_2(\delta)$. Its first level is the two-row hyperoctahedral representation graph associated with type $S^{(n-k,k)}$. Retaining every two-row irreducible and every coordinate box-transfer channel, together with a moving-projection theorem, yields an explicit four-parameter exponent $\kappa_{\mathrm{HC}}$. The earlier whole-cube exponent $\kappa_H$ is a boundary restriction of this optimization, whereas the fully optimized second MRRW exponent $M_2$ is an exact symmetric slice. The prior best curve is the combined $\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$, which uses a constant-weight branch $\kappa_{\mathrm{CW}}$. Replacing only the whole-cube branch by the honeycomb bound gives $\kappa_{\mathrm{best}}=\min\{\kappa_{\mathrm{CW}}, \kappa_{\mathrm{HC}}\}$. We prove, on $0<\delta<1/2$, \[ R_2(\delta)\le \kappa_{\mathrm{best}}(\delta) \le \kappa_{\mathrm{bin}}(\delta) \le R_{\mathrm{2MQC}}(\delta)<M_2(\delta),\\[-1mm] \kappa_{\mathrm{best}}(\delta) \le \min\{\kappa_{\mathrm{CW}}(\delta), \kappa_{\mathrm{bal}}(\delta)\} <R_{\mathrm{2MQC}}(\delta), \qquad \kappa_H(\delta)=R_{\mathrm{MQC}}(\delta). \] The hierarchy has two further directions. Increasing the representation depth replaces scalar by matrix-valued transfers on the hive. Increasing the anchor depth localizes it in a stable-set hierarchy. The resulting bounds are monotone in both directions and eventually recover $A_2(n,d)$. A complementary Horn--channel hierarchy gives matrix optimizations whose $2\times2$ level is $\kappa_{\mathrm{HC}}$ and whose $3\times3$ level is a stronger bound. Already at low levels, they can be used to improve the strongest previous general bounds, while the honeycomb framework provides a route towards tighter bounds.

cs.IT↗

arXiv2026

Dimension-Free Polylogarithmic Quantum Shadow Tomography from Sequential Pretty-Good Measurements

\textit{Shadow tomography} is a fundamental problem in quantum information theory. Given multiple copies of an unknown $d$-dimensional quantum state $\rho$ and a known collection of observables ${E_1,\ldots,E_m}$, the goal is to estimate all expectation values $\{\Tr(\rho E_i)\}_{i=1}^m$ to additive accuracy $\varepsilon$ with probability at least $1-\delta$. An elusive open question from the seminal shadow tomography work of Aaronson (STOC'18) is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on $m$, as suggested by the best-known lower bounds. In this work, we give a quantum protocol for shadow tomography with sample complexity \[ O\left( \frac{1}{\varepsilon^2} \frac{(\log (m/\delta))^4} {(\log\log (m/\delta))^3} \right), \] which is polylogarithmic in the number of observables and independent of the dimension of the unknown state thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography from Sinha (STOC'25). Our approach first reduces the general shadow-tomography problem to a finite-ensemble estimation problem via a minimax argument. We then develop an observable-independent protocol that repeatedly applies the pretty-good measurement and updates the priori distribution over the finite ensemble according to the measurement outcomes. A refined tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables.

quant-ph↗

arXiv2026

$(5+\epsilon)$-Approximation of Fr\'echet Distance in Strongly Subquadratic Time

We give randomized $(5+\epsilon)$-approximation algorithms for both the continuous and discrete Fr\'echet distances on arbitrary two polygonal curves $\tau$ and $\sigma$ in $\mathbb R^d$ for fixed $d$, with $n$ and $m\le n$ vertices respectively. Our algorithm for continuous Fr\'echet runs in $\widetilde O_{d,\epsilon}(n m^{8/9})$ time, and our algorithm for discrete Fr\'echet runs in $\widetilde O_{d,\epsilon}(n m^{4/5})$ time. These bounds improve the recent strongly subquadratic constant-factor approximation algorithms of Cheng, Huang, and Zhang~\cite{cheng2025constant}, which give $(7+\epsilon)$-approximations. The approximation improvement comes from certifying long boundary-to-boundary reachability directly through auxiliary surrogate curves, avoiding an extra conversion back to input subcurves and hence removing one triangle-inequality loss. The running-time improvement comes from a two-scale macro-surrogate search combined with dyadic auxiliary-transfer structures, with the discrete case gaining a faster bound from exact planar reachability in the discrete free-space graph.

cs.CG↗

arXiv2026

Optimal Proximity Gap for Folded Reed--Solomon Codes via Subspace Designs

A collection of sets satisfies a $(\delta,\varepsilon)$-proximity gap with respect to some property if for every set in the collection, either (i) all members of the set are $\delta$-close to the property in (relative) Hamming distance, or (ii) only a small $\varepsilon$-fraction of members are $\delta$-close to the property. In a seminal work, Ben-Sasson \textit{et al.}\ showed that the collection of affine subspaces exhibits a $(\delta,\varepsilon)$-proximity gap with respect to the property of being Reed--Solomon (RS) codewords with $\delta$ up to the so-called Johnson bound for list decoding. Their technique relies on the Guruswami--Sudan list decoding algorithm for RS codes, which is guaranteed to work in the Johnson bound regime. Folded Reed--Solomon (FRS) codes are known to achieve the optimal list decoding radius $\delta$, a regime known as capacity. Moreover, a rich line of list decoding algorithms was developed for FRS codes. It is then natural to ask if FRS codes can be shown to exhibit an analogous $(\delta,\varepsilon)$-proximity gap, but up to the so-called optimal capacity regime. We answer this question in the affirmative (and the framework naturally applies more generally to suitable subspace-design codes). An additional motivation to understand proximity gaps for FRS codes is the recent results [BCDZ'25] showing that they exhibit properties similar to random linear codes, which were previously shown to be related to properties of RS codes with random evaluation points in [LMS'25], as well as codes over constant-size alphabet based on AEL [JS'25].

cs.IT↗