arXiv subjects

John Wright

Publications and source records attributed to John Wright.

At least 19 recordsLinked to original sources

arXiv2026

Online Shadow Tomography Matching the Classical Bounds

In Online Shadow Tomography, we are given copies of an unknown $d$-dimensional quantum state $\rho$, an adversary (adaptively) proposes a sequence of bounded observables $A^{(1)},\ldots,A^{(m)}$, and after each $A^{(t)}$ is given we must estimate $\mathrm{Tr}(A^{(t)}\rho)$ to within $\pm \epsilon$. This is the direct quantum generalization of the classical problem of Adaptive Data Analysis. Prior results for online Shadow Tomography were suboptimal in all three parameters $m, d, \epsilon$, lagging behind the best known and classical rates, for which there is some evidence of optimality. In this work, we finally close this gap, giving a pair of algorithms matching the classical rates. Our first algorithm is the first to achieve $o(\log^2 m)$-dependence together with $\mathrm{poly}(\log(d)/\epsilon)$; moreover, it improves all three exponents even in the Offline Shadow Tomography setting. Our second algorithm is known to be optimal among bounds independent of $d$, and improves the best prior result by a $\sqrt{m} \log m$ factor. The key to our proof is a new framework for quantifying post-measurement damage, based on the quantum Efron-Stein decomposition.

quant-ph↗

arXiv2026

No low-degree tests for quantum states

We study the problem of testing low-degree phase states, namely m-qudit quantum states of the form $q^{-m/2} \sum_{x \in \mathbb{F}_q^m} \omega^{f(x)} |x>$, where $f$ is a degree-$d$ polynomial. In contrast to the classical setting, where low-degree polynomials admit highly efficient classical testers, it is not known whether analogous quantum tests exist. We show that no such quantum low-degree test exists: any tester requires $\Omega(\binom{\lfloor m/2\rfloor}{\lfloor (d-1)/2 \rfloor})$ copies to determine whether a given state is a degree-$d$ phase state or is far from every such state. Our results follow from a general framework that relates quantum testing of codeword phase states to classical decoding properties of the dual code, which allows us to leverage known bounds on the tolerance of high-rate Reed--Muller codes to random errors.

quant-ph↗

arXiv2026

The Keyl-Werner algorithm is not optimal for spectrum estimation

We give an algorithm which, given $n = O(d^2 \cdot (\log\log(d)/\log(d))^2)$ copies of $\rho$, estimates the eigenvalues of $\rho$ to constant error in total variation distance. Thus, we can learn the eigenvalues of a quantum state with fewer copies than the $\Theta(d^2)$ needed to run full state tomography. This is the first improvement to spectrum estimation over the influential Keyl-Werner algorithm, which uses $n = \Theta(d^2)$ copies, thereby resolving a question raised by Keyl and Werner in 2001 and refuting a 2016 conjecture of Wright. Our main technical tool is a new tomography guarantee, where the error of tomography in a particular direction $|w\rangle$ scales with $\langle w | \rho |w\rangle$ for all directions simultaneously. From this stronger "relative-error" bound, we recover better algorithms for principal component analysis in Bures distance and tomography in $\chi^2$-divergence as corollaries.

quant-ph↗

arXiv2026

Quantum Lazy Sampling and Path Recording for Any Group

A central challenge in quantum algorithms and cryptography is reasoning about algorithms with oracle access to a random group element (e.g. a random function, permutation, or unitary). Can we efficiently simulate such algorithms? Can we determine what they know after t queries? A classical tool for this is lazy sampling: the oracle does not commit to the full group element upfront, but rather samples partial information about it on the fly. We study a quantum analog of lazy sampling: compressed oracles (or recording oracles). These are quantum data structures that allow on-the-fly simulation for quantum queries, originally introduced by Zhandry (CRYPTO '19) for random functions, and generalized to unitaries by Ma-Huang (STOC '25) and permutations by Carolan (STOC '26), and used to great effect in security proofs and lower bounds due to their interpretability. We define and analyze a general-purpose and interpretable path-recording oracle, derived from first principles, that perfectly simulates random elements of any closed subgroup of $U(N)$. Our oracle stores, in superposition, t input-output pairs, with updates described in terms of the commutant of the group's tensor power representation. This transparently records the information the algorithm has learned. Our oracle builds on recent work of Grinko-Yoshida (QIP '26), who gave a different general-purpose compressed oracle without clear interpretability. One interesting application of our path-recording is allowing direct comparisons between compressed oracles of different groups, giving a new technique for proving pseudorandomness results. For example, comparing $S_N$ and $U(N)$ yields what is arguably the simplest construction to date of pseudorandom unitaries: the product PC of a pseudorandom permutation and a random Clifford, improving on the prior PFC construction (Metger-Poremba-Sinha-Yuen, FOCS '24; Ma-Huang, STOC '25).

quant-ph↗

arXiv2026

Learning the structure of open quantum systems

We design an algorithm for learning the coefficients of an $n$-qubit constant-local Lindbladian to $\varepsilon$ error with $O(g d^2 \log(n) / \varepsilon^2)$ total evolution time, where $g$ is the single-site energy and $d$ is the (approximate) degree of the interaction graph. Though Lindbladians present new challenges not present in the special case of Hamiltonians, our algorithm achieves the suite of desiderata attained by state-of-the-art Hamiltonian learning algorithms: (1) it uses non-adaptive, ancilla-free randomized Pauli measurement circuits with a time resolution of only $\Theta(1/g)$; (2) it works without knowledge of the structure of the unknown Lindbladian; (3) it depends on a smooth form of degree, thereby supporting the learning of quasi-local and power-law Lindbladians. Our algorithm is a simple iterative method, where the objective function consists of Fourier coefficients of the Lindbladian restricted to few-site regions. Its analysis identifies the difficulty unique to open systems, which we call "confusing" terms. For settings where the "confusion" is limited, the performance of the algorithm improves. We demonstrate this for the case of structure learning of Hamiltonians from access to real-time evolution, where we obtain a new algorithm that is significantly simpler than previous work. In addition, using the same iterative method, we design the first efficient algorithm for structure learning Hamiltonians from high-temperature Gibbs states.

quant-ph↗

arXiv2026

Nonasymptotic bounds for quantum purity amplification

In quantum purity amplification, one is given $n$ copies of a noisy quantum state $\rho \in \mathbb{C}^{d \times d}$ and asked to prepare $k$ copies of its principal eigenstate $|v_d\rangle$. Several prior works have derived information-theoretically optimal algorithms for this problem, but the bounds they prove are only shown in the asymptotic regime as the number of samples $n$ tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if $\rho$'s eigenvalues are sorted $p_1 \leq \cdots \leq p_d$ and $p_{d-1} < p_d$, then \begin{equation*} n = O\Big(k + \frac{k}{\delta} \cdot \frac{1-p_d}{(p_d-p_{d-1})^2}\Big) \end{equation*} copies suffice to output a state with fidelity at least $1-\delta$ with $|v_d^{\otimes k}\rangle$. Our bound holds for arbitrary spectra, and is independent of the dimension $d$. In the case of depolarizing noise, our finite-sample guarantee matches the optimal asymptotic scaling. Our proof is based on the combinatorics of random Young diagrams.

quant-ph↗

arXiv2026

How Fusion-Born Alpha Particles Suppress Microturbulence in Burning Plasmas

A central unresolved question in fusion energy research is whether energetic alpha particles, the primary products of deuterium-tritium fusion reactions, enhance or degrade plasma confinement. In burning plasmas, the operating regime of future devices such as ITER and SPARC, alpha particles become the dominant heating source, yet their impact on confinement has remained uncertain. Here, we present self-consistent simulations of burning plasmas that simultaneously evolve microturbulence, alpha-particle heating, and macroscopic plasma profiles to steady state, and find that alpha particles can substantially improve confinement. Fusion-born alpha particles weakly destabilize toroidal Alfven eigenmodes (TAEs), which nonlinearly enhance zonal flows that shear apart and suppress ion-scale turbulence. The resulting reduction in turbulent heat transport drives stronger core profile peaking, increasing alpha heating by up to 25% and establishing a self-reinforcing feedback loop. This mechanism has no direct analogue in present-day experiments, where external heating dominates, and reveals an intrinsic pathway toward improved confinement in burning plasmas.

physics.plasm-ph↗

arXiv2025

Improved Lower Bounds for QAC0

In this work, we prove the strongest known lower bounds for QAC$^0$, allowing polynomially many gates and ancillae. Our main results show that: (1) Depth-3 QAC$^0$ circuits cannot compute PARITY, and require $\Omega(\exp(\sqrt{n}))$ gates to compute MAJORITY. (2) Depth-2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage, regardless of size. We develop new classical simulation techniques for QAC$^0$ to obtain our depth-3 bounds. In these results, we relax the output requirement of the quantum circuit to a single bit, making our depth $2$ approximation bound stronger than the previous best bound of Rosenthal (2021). This also enables us to draw natural comparisons with classical AC$^0$ circuits, which can compute PARITY exactly in depth $2$ (exp size). Our techniques further suggest that, for boolean total functions, constant-depth quantum circuits do not necessarily provide more power than their classical counterparts. Our third result shows that depth $2$ QAC$^0$ circuits, regardless of size, cannot exactly synthesize an $n$-target nekomata state (a state whose synthesis is directly related to the computation of PARITY). This complements the depth $2$ exponential size upper bound of Rosenthal (2021) for approximating nekomatas (which is used as a sub-circuit in the only known constant depth PARITY upper bound). Finally, we argue that approximating PARITY in QAC0, with significantly better than 1/poly(n) advantage on average, is just as hard as computing it exactly. Thus, extending our techniques to higher depths would also rule out approximate circuits for PARITY and related problems

quant-ph↗

arXiv2025

Mixed state tomography reduces to pure state tomography

A longstanding belief in quantum tomography is that estimating a mixed state is far harder than estimating a pure state. This is borne out in the mathematics, where mixed state algorithms have always required more sophisticated techniques to design and analyze than pure state algorithms. We present a new approach to tomography demonstrating that, contrary to this belief, state-of-the-art mixed state tomography follows easily and naturally from pure state algorithms. We analyze the following strategy: given $n$ copies of an unknown state $\rho$, convert them into copies of a purification $|\rho\rangle$; run a pure state tomography algorithm to produce an estimate of $|\rho\rangle$; and output the resulting estimate of $\rho$. The purification subroutine was recently discovered via the "acorn trick" of Tang, Wright, and Zhandry. With this strategy, we obtain the first tomography algorithm which is sample-optimal in all parameters. For a rank-$r$ $d$-dimensional state, it uses $n = O((rd + \log(1/\delta))/\varepsilon)$ samples to output an estimate which is $\varepsilon$-close in fidelity with probability at least $1-\delta$. This algorithm also uses poly$(n)$ gates, making it the first gate-efficient tomography algorithm which is sample-optimal even in terms of the dimension $d$ alone. Moreover, with this method we recover essentially all results on mixed state tomography, including its applications to tomography with limited entanglement, classical shadows, and quantum metrology. Our proofs are simple, closing the gap in conceptual difficulty between mixed and pure tomography. Our results also clarify the role of entangled measurement in mixed state tomography: the only step of the algorithm which requires entanglement across copies is the purification step, suggesting that, for tomography, the reason entanglement is useful is for consistent purification.

quant-ph↗

arXiv2025

Optimal lower bounds for quantum state tomography

We show that $n = \Omega(rd/\varepsilon^2)$ copies are necessary to learn a rank $r$ mixed state $\rho \in \mathbb{C}^{d \times d}$ up to error $\varepsilon$ in trace distance. This matches the upper bound of $n = O(rd/\varepsilon^2)$ from prior work, and therefore settles the sample complexity of mixed state tomography. We prove this lower bound by studying a special case of full state tomography that we refer to as projector tomography, in which $\rho$ is promised to be of the form $\rho = P/r$, where $P \in \mathbb{C}^{d \times d}$ is a rank $r$ projector. A key technical ingredient in our proof, which may be of independent interest, is a reduction which converts any algorithm for projector tomography which learns to error $\varepsilon$ in trace distance to an algorithm which learns to error $O(\varepsilon)$ in the more stringent Bures distance.

quant-ph↗

arXiv2025

The debiased Keyl's algorithm: a new unbiased estimator for full state tomography

In the problem of quantum state tomography, one is given $n$ copies of an unknown rank-$r$ mixed state $\rho \in \mathbb{C}^{d \times d}$ and asked to produce an estimator of $\rho$. In this work, we present the debiased Keyl's algorithm, the first estimator for full state tomography which is both unbiased and sample-optimal. We derive an explicit formula for the second moment of our estimator, with which we show the following applications. (1) We give a new proof that $n = O(rd/\varepsilon^2)$ copies are sufficient to learn a rank-$r$ mixed state to trace distance error $\varepsilon$, which is optimal. (2) We further show that $n = O(rd/\varepsilon^2)$ copies are sufficient to learn to error $\varepsilon$ in the more challenging Bures distance, which is also optimal. (3) We consider full state tomography when one is only allowed to measure $k$ copies at once. We show that $n =O\left(\max \left(\frac{d^3}{\sqrt{k}\varepsilon^2}, \frac{d^2}{\varepsilon^2} \right) \right)$ copies suffice to learn in trace distance. This improves on the prior work of Chen et al. and matches their lower bound. (4) For shadow tomography, we show that $O(\log(m)/\varepsilon^2)$ copies are sufficient to learn $m$ given observables $O_1, \dots, O_m$ in the "high accuracy regime", when $\varepsilon = O(1/d)$, improving on a result of Chen et al. More generally, we show that if $\mathrm{tr}(O_i^2) \leq F$ for all $i$, then $n = O\Big(\log(m) \cdot \Big(\min\Big\{\frac{\sqrt{r F}}{\varepsilon}, \frac{F^{2/3}}{\varepsilon^{4/3}}\Big\} + \frac{1}{\varepsilon^2}\Big)\Big)$ copies suffice, improving on existing work. (5) For quantum metrology, we give a locally unbiased algorithm whose mean squared error matrix is upper bounded by twice the inverse of the quantum Fisher information matrix in the asymptotic limit of large $n$, which is optimal.

quant-ph↗

arXiv2025

Haar random codes attain the quantum Hamming bound, approximately

We study the error correcting properties of Haar random codes, in which a $K$-dimensional code space $\boldsymbol{C} \subseteq \mathbb{C}^N$ is chosen at random from the Haar distribution. Our main result is that Haar random codes can approximately correct errors up to the quantum Hamming bound, meaning that a set of $m$ Pauli errors can be approximately corrected so long as $mK \ll N$. This is the strongest bound known for any family of quantum error correcting codes (QECs), and continues a line of work showing that approximate QECs can significantly outperform exact QECs [LNCY97, CGS05, BGG24]. Our proof relies on a recent matrix concentration result of Bandeira, Boedihardjo, and van Handel.

quant-ph↗

arXiv2025

Conjugate queries can help

We give a natural problem over input quantum oracles $U$ which cannot be solved with exponentially many black-box queries to $U$ and $U^\dagger$, but which can be solved with constant many queries to $U$ and $U^*$, or $U$ and $U^{\mathrm{T}}$. We also demonstrate a quantum commitment scheme that is secure against adversaries that query only $U$ and $U^\dagger$, but is insecure if the adversary can query $U^*$. These results show that conjugate and transpose queries do give more power to quantum algorithms, lending credence to the idea put forth by Zhandry that cryptographic primitives should prove security against these forms of queries. Our key lemma is that any circuit using $q$ forward and inverse queries to a state preparation unitary for a state $\sigma$ can be simulated to $\varepsilon$ error with $n = \mathcal{O}(q^2/\varepsilon)$ copies of $\sigma$. Consequently, for decision tasks, algorithms using (forward and inverse) state preparation queries only ever perform quadratically better than sample access. We also identify a motif, which we call the "acorn trick", where generically strengthening a quantum resource can be possible if the output is allowed to be random, bypassing no-go theorems for deterministic algorithms. We demonstrate this idea for several settings, including controlization and purification.

quant-ph↗

arXiv2025

Amplitude amplification and estimation require inverses

We prove that the generic quantum speedups for brute-force search and counting only hold when the process we apply them to can be efficiently inverted. The algorithms speeding up these problems, amplitude amplification and amplitude estimation, assume the ability to apply a state preparation unitary $U$ and its inverse $U^\dagger$; we give problem instances based on trace estimation where no algorithm which uses only $U$ beats the naive, quadratically slower approach. Our proof of this is simple and goes through the compressed oracle method introduced by Zhandry. Since these two subroutines are responsible for the ubiquity of the quadratic "Grover" speedup in quantum algorithms, our result explains why such speedups are far harder to come by in the settings of quantum learning, metrology, and sensing. In these settings, $U$ models the evolution of an experimental system, so implementing $U^\dagger$ can be much harder -- tantamount to reversing time within the system. Our result suggests a dichotomy: without inverse access, quantum speedups are scarce; with it, quantum speedups abound.

quant-ph↗

arXiv2025

Are controlled unitaries helpful?

Many quantum algorithms, to compute some property of a unitary $U$, require access not just to $U$, but to $cU$, the unitary with a control qubit. We show that having access to $cU$ does not help for a large class of quantum problems. For a quantum circuit which uses $cU$ and $cU^\dagger$ and outputs $|\psi(U)\rangle$, we show how to "decontrol" the circuit into one which uses only $U$ and $U^\dagger$ and outputs $|\psi(\varphi U)\rangle$ for a uniformly random phase $\varphi$, with a small amount of time and space overhead. When we only care about the output state up to a global phase on $U$, then the decontrolled circuit suffices. Stated differently, $cU$ is only helpful because it contains global phase information about $U$. A version of our procedure is described in an appendix of Sheridan, Maslov, and Mosca (arXiv:0810.3843). Our goal with this work is to popularize this result by generalizing it and investigating its implications, in order to counter negative results in the literature which might lead one to believe that decontrolling is not possible. As an application, we give a simple proof for the existence of unitary ensembles which are pseudorandom under access to $U$, $U^\dagger$, $cU$, and $cU^\dagger$.

quant-ph↗

arXiv2025

Local Averaging Accurately Distills Manifold Structure From Noisy Data

High-dimensional data are ubiquitous, with examples ranging from natural images to scientific datasets, and often reside near low-dimensional manifolds. Leveraging this geometric structure is vital for downstream tasks, including signal denoising, reconstruction, and generation. However, in practice, the manifold is typically unknown and only noisy samples are available. A fundamental approach to uncovering the manifold structure is local averaging, which is a cornerstone of state-of-the-art provable methods for manifold fitting and denoising. However, to the best of our knowledge, there are no works that rigorously analyze the accuracy of local averaging in a manifold setting in high-noise regimes. In this work, we provide theoretical analyses of a two-round mini-batch local averaging method applied to noisy samples drawn from a $d$-dimensional manifold $\mathcal M \subset \mathbb{R}^D$, under a relatively high-noise regime where the noise size is comparable to the reach $\tau$. We show that with high probability, the averaged point $\hat{\mathbf q}$ achieves the bound $d(\hat{\mathbf q}, \mathcal M) \leq \sigma \sqrt{d\left(1+\frac{\kappa\mathrm{diam}(\mathcal {M})}{\log(D)}\right)}$, where $\sigma, \mathrm{diam(\mathcal M)},\kappa$ denote the standard deviation of the Gaussian noise, manifold's diameter and a bound on its extrinsic curvature, respectively. This is the first analysis of local averaging accuracy over the manifold in the relatively high noise regime where $\sigma \sqrt{D} \approx \tau$. The proposed method can serve as a preprocessing step for a wide range of provable methods designed for lower-noise regimes. Additionally, our framework can provide a theoretical foundation for a broad spectrum of denoising and dimensionality reduction methods that rely on local averaging techniques.

stat.ML↗

arXiv2025

Beating full state tomography for unentangled spectrum estimation

How many copies of a mixed state $\rho \in \mathbb{C}^{d \times d}$ are needed to learn its spectrum? To date, the best known algorithms for spectrum estimation require as many copies as full state tomography, suggesting the possibility that learning a state's spectrum might be as difficult as learning the entire state. We show that this is not the case in the setting of unentangled measurements, by giving a spectrum estimation algorithm that uses $n = O(d^3\cdot (\log\log(d) / \log(d))^4 )$ copies of $\rho$, which is asymptotically fewer than the $n = \Omega(d^3)$ copies necessary for full state tomography. Our algorithm is inspired by the technique of local moment matching from classical statistics, and shows how it can be applied in the quantum setting. As an important subroutine in our spectrum estimation algorithm, we give an estimator of the $k$-th moment $\operatorname{tr}(\rho^k)$ which performs unentangled measurements and uses $O(d^{3-2/k})$ copies of $\rho$ in order to achieve a constant multiplicative error. This directly translates to an additive-error estimator of quantum Renyi entropy of order $k$ with the same number of copies. Finally, we present numerical evidence that the sample complexity of spectrum estimation can only improve over full state tomography by a sub-polynomial factor. Specifically, for spectrum learning with fully entangled measurements, we run simulations which suggest a lower bound of $\Omega(d^{2 - \gamma})$ copies for any constant $\gamma > 0$. From this, we conclude the current best lower bound of $\Omega(d)$ is likely not tight.

quant-ph↗

arXiv2025

Growth and characterization of single crystal cubic TaN and hexagonal Ta$_2$N films on c-plane Sapphire

Two single crystal phases of tantalum nitride were stabilized on c-plane sapphire using molecular beam epitaxy. The phases were identified to be $\delta$-TaN with a rocksalt cubic structure and $\gamma$-Ta$_2$N with a hexagonal structure. Atomic force microscopy scans revealed smooth surfaces for both the films with root mean square roughnesses less than 0.3 nm. Phase-purity of these films was determined by x-ray diffraction. Raman spectrum of the phase-pure $\delta$-TaN and $\gamma$-Ta$_2$N obtained will serve as a future reference to determine phase-purity of tantalum nitride films. Further, the room-temperature and low-temperature electronic transport measurements indicated that both of these phases are metallic at room temperature with resistivities of 586.2 $\mu\Omega$-cm for the 30 nm $\delta$-TaN film and 75.5 $\mu\Omega$-cm for the 38 nm $\gamma$-Ta$_2$N film and become superconducting below 3.6 K and 0.48 K respectively. The superconducting transition temperature reduces with applied magnetic field as expected. Ginzburg-Landau fitting revealed a 0 K critical magnetic field and coherence length of 18 T and 4.2 nm for the 30 nm $\delta$-TaN film and 96 mT and 59 nm for the 38 nm $\gamma$-Ta$_2$N film. These tantalum nitride films are of high interest for superconducting resonators and qubits.

cond-mat.supr-con↗