Fermionic tomography and learning

• URL: http://arxiv.org/abs/2207.14787v1
• Date: Fri, 29 Jul 2022 17:12:53 GMT
• Title: Fermionic tomography and learning
• Authors: Bryan O'Gorman
• Abstract summary: Shadow tomography via classical shadows is a state-of-the-art approach for estimating properties of a quantum state. We show how this approach leads to efficient estimation protocols for the fidelity with a pure fermionic Gaussian state. We use these tools to show that an $n$-electron, $m$-mode Slater can be learned to within $epsilon$ fidelity given $O(n2 m7 log(m / delta) / epsilon2)$ samples of the determinant.
• Score: 0.5482532589225553
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Shadow tomography via classical shadows is a state-of-the-art approach for estimating properties of a quantum state. We present a simplified, combinatorial analysis of a recently proposed instantiation of this approach based on the ensemble of unitaries that are both fermionic Gaussian and Clifford. Using this analysis, we derive a corrected expression for the variance of the estimator. We then show how this leads to efficient estimation protocols for the fidelity with a pure fermionic Gaussian state (provably) and for an $X$-like operator of the form ($|\mathbf 0\rangle\langle\psi|$ + h.c.) (via numerical evidence). We also construct much smaller ensembles of measurement bases that yield the exact same quantum channel, which may help with compilation. We use these tools to show that an $n$-electron, $m$-mode Slater determinant can be learned to within $\epsilon$ fidelity given $O(n^2 m^7 \log(m / \delta) / \epsilon^2)$ samples of the Slater determinant.

Related papers

• Resource-efficient shadow tomography using equatorial stabilizer measurements [0.0]
  equatorial-stabilizer-based shadow-tomography schemes can estimate $M$ observables using $mathcalO(log(M),mathrmpoly(n),1/varepsilon2)$ sampling copies. We numerically confirm our theoretically-derived shadow-tomographic sampling complexities with random pure states and multiqubit graph states.
  arXiv  Detail & Related papers  (2023-11-24T17:33:44Z)
• Quantum Simulation of the First-Quantized Pauli-Fierz Hamiltonian [0.5097809301149342]
  We show that a na"ive partitioning and low-order splitting formula can yield, through our divide and conquer formalism, superior scaling to qubitization for large $Lambda$. We also give new algorithmic and circuit level techniques for gate optimization including a new way of implementing a group of multi-controlled-X gates.
  arXiv  Detail & Related papers  (2023-06-19T23:20:30Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• Quantum tomography using state-preparation unitaries [0.22940141855172028]
  We describe algorithms to obtain an approximate classical description of a $d$-dimensional quantum state when given access to a unitary. We show that it takes $widetildeTheta(d/varepsilon)$ applications of the unitary to obtain an $varepsilon$-$ell$-approximation of the state. We give an efficient algorithm for obtaining Schatten $q$-optimal estimates of a rank-$r$ mixed state.
  arXiv  Detail & Related papers  (2022-07-18T17:56:18Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• $O(N^2)$ Universal Antisymmetry in Fermionic Neural Networks [107.86545461433616]
  We propose permutation-equivariant architectures, on which a determinant Slater is applied to induce antisymmetry. FermiNet is proved to have universal approximation capability with a single determinant, namely, it suffices to represent any antisymmetric function. We substitute the Slater with a pairwise antisymmetry construction, which is easy to implement and can reduce the computational cost to $O(N2)$.
  arXiv  Detail & Related papers  (2022-05-26T07:44:54Z)
• Fermionic partial tomography via classical shadows [0.0]
  We propose a tomographic protocol for estimating any $ k $-body reduced density matrix ($ k $-RDM) of an $ n $-mode fermionic state. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum-state properties, to the fermionic setting.
  arXiv  Detail & Related papers  (2020-10-30T06:28:26Z)
• Bosonic and fermionic Gaussian states from K\"ahler structures [0.0]
  We show that bosonic and fermionic Gaussian states can be uniquely characterized by their linear complex structure $J$. We apply these methods to the study of entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of driven systems.
  arXiv  Detail & Related papers  (2020-10-29T12:21:47Z)
• Local optimization on pure Gaussian state manifolds [63.76263875368856]
  We exploit insights into the geometry of bosonic and fermionic Gaussian states to develop an efficient local optimization algorithm. The method is based on notions of descent gradient attuned to the local geometry. We use the presented methods to collect numerical and analytical evidence for the conjecture that Gaussian purifications are sufficient to compute the entanglement of purification of arbitrary mixed Gaussian states.
  arXiv  Detail & Related papers  (2020-09-24T18:00:36Z)
• Spectral density estimation with the Gaussian Integral Transform [91.3755431537592]
  spectral density operator $hatrho(omega)=delta(omega-hatH)$ plays a central role in linear response theory. We describe a near optimal quantum algorithm providing an approximation to the spectral density.
  arXiv  Detail & Related papers  (2020-04-10T03:14:38Z)
• Quantum Algorithms for Simulating the Lattice Schwinger Model [63.18141027763459]
  We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In lattice units, we find a Schwinger model on $N/2$ physical sites with coupling constant $x-1/2$ and electric field cutoff $x-1/2Lambda$. We estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable---the mean pair density.
  arXiv  Detail & Related papers  (2020-02-25T19:18:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.