Learning compact, interpretable descriptions of quantum many-body states is an important task in quantum science. We study the task of learning the Slater determinant with maximum fidelity to an arbitrary fermionic many-body state, with motivation from both Hartree-Fock methods and agnostic tomography. Given an $n$-fermion wavefunction built from $m$ fermionic modes, we provide classical and quantum algorithms returning a Slater determinant with fidelity within $\varepsilon$ of maximal in time $m^{\text{poly}(n,1/\varepsilon)}$. We prove matching hardness lower bounds, assuming standard complexity conjectures, along some parameter axes. Given access to quantum copies, we prove this can be accomplished with $\text{poly}(m,n,1/\varepsilon)$ copies of $\rho$. We also show that above a fidelity of $2/3$ any stationary point is the unique global maximum while below $2/3$ the optimization landscape can have spurious stationary points, and hence $2/3$ marks a transition point in the optimization landscape for this problem. We apply the algorithm to the Fermi-Hubbard model, extracting the closest Slater determinant from neural quantum state solutions. Together, our results provide algorithmic tools with provable guarantees in understanding fermionic many-body systems with classical or quantum simulation.
Deep parameterized quantum circuits may remain sensitive to a parameter change while the observables retained by a learning model barely respond. We study this separation for a fixed computational-basis measurement. For a pure-state tangent, we compare the quantum Fisher information $F_Q$, the Fisher information $F_{\rm full}$ in the complete bitstring distribution, and the largest variance-normalized response $\mathcal I_{\mathcal A}$ available to a diagonal readout space $\mathcal A$. If the joint state--tangent frame is Haar random, we prove that the two successive information fractions are independent Beta variables whose means are $1/2$ and $r/(2^n-1)$, where $r$ is the centered dimension of the readout. Consequently, even the joint span of all computational-basis Pauli strings through any fixed weight $k$ retain only $O(n^k2^{-n})$ of the full-record information. Exact-statevector experiments across six circuit families show increasing finite-size agreement with this hierarchy in five nonconserving ensembles as the circuit depth grows. A number-conserving family departs strongly from the isotropic prediction even after correcting the support and readout rank, showing that rank alone is insufficient without tangent isotropy.
We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize $\langle(x \otimes y), M (x \otimes y)\rangle$ over unit vectors $x,y$ where $0 \preceq M \preceq I$; we call this value $\mathrm{BSS}(M)$. We study $\mathrm{BSS}$ in the"perfect completeness"regime, where given $M$ such that $\mathrm{BSS}(M) = 1$ the goal is to find the best possible solution $x,y$ -- this generalizes the problem of finding a rank-one matrix as close as possible to a given subspace of $\mathbb{R}^{n \times n}$ guaranteed to contain a rank-one matrix. The strongest known algorithmic guarantees for this problem are: (1) an algorithm which finds a solution with value $1-\varepsilon$ in time $\exp(\sqrt{n} (\log n)^{O(1)} / \varepsilon^2)$, due to Barak, Kothari, and Steurer, and (2) an algorithm which finds a solution with value $q/n$ in time roughly $n^{O(q)}$, due to Bhattiprolu, Ghosh, Guruswami, Lee, and Tulsiani. We give a much simpler approach to rounding the SoS relaxation, generalizing the canonical"global correlation rounding"technique, and obtain a better running time. Given $M$ with $\mathrm{BSS}(M) = 1$, our algorithm finds a solution with value $1-\epsilon$ in time $n^{O(\sqrt{n/\varepsilon})}$, and a solution of value $q/n$ in time $n^{O(\sqrt q)}$. Using the same techniques, we prove a new variant of the"pinning lemma", a measure-decomposition theorem widely used in LP/SDP rounding, high-dimensional probability, and statistical physics, which we believe is of independent interest.
Prashanti Anderson, Sam Hopkins, Amit Rajaraman· 1 citation
\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.
Efficient, deterministic, and high-fidelity preparation of large Fock states is essential for scaling bosonic quantum technologies and exploring quantum phenomena at large excitation energies. We introduce a deterministic one-parameter (D1p) protocol that maps Fock-state preparation in an infinite-dimensional Hilbert space onto two-dimensional amplitude amplification. Starting from a coherent state with $|\alpha|\simeq\sqrt{n}$, the initial target-state population scales as $n^{-1/2}$, yielding an iteration count and circuit depth of $\mathcal{O}(n^{1/4})$. Phase matching guarantees unit fidelity in the ideal model; remarkably, preparing $|{10^6}\rangle$ requires only 39 iterations. The protocol uses only displacements and number-selective phase operations, requires no numerical optimization, and further extends to state transfer, general superpositions, finite-dimensional systems, and multipartite entangled states. In the large-amplitude regime, its multi-target form prepares $L$-legged cat states with an iteration count determined only by $L$; cats with up to ten legs require only two iterations, independent of the coherent-state amplitude. This framework provides a broadly applicable route to highly excited bosonic states on platforms supporting these elementary controls.
The unitary brick-wall is proposed: a $k-particle fermionic architecture for nearest-neighbor hardware, combining Reconfigurable Beam Splitter gates with interleaved single-qubit phase gates and a non-Gaussian magic-state encoding.
This paper invalidates the algorithm-relative exponential-cost conclusion for the supervised two-body readout, without affecting the gradient-variance, barren-plateau, parameter-shift, or sampling-hardness results.
Erfan Amidi· 0 citations
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