It is shown how this algorithm is able to take advantage of emerging dynamical circuit capabilities in near-term hardware to roughly halve the number required qubits, as well as how quantum readout error mitigation is trivial for this method.
Abstract
Maximally mixed state spectral sampling is an unbiased quantum algorithm that allows for extraction of a finite-resolution spectral distribution from a Hamiltonian over potentially the entire allowed range of energies. We show how it may be focused on any desired area of the spectrum in order to learn about the full \textit{fingerprint} of the model of interest: from its ground state phenomena such as quantum criticality, obtained from the lowest lying energies, to its thermalization behavior, obtained from the mid-spectrum. We demonstrate this technique specifically on a $(1+1)d$ non-Abelian $SO(3)$ gauge theory, providing a comprehensive analysis of the steps necessary for performing this algorithm, as well as what is possible in the near-term with superconducting quantum hardware, performing simulations with circuits that are $78$ two-qubit gates deep. We show how this algorithm is able to take advantage of emerging dynamical circuit capabilities in near-term hardware to roughly halve the number required qubits, as well as how quantum readout error mitigation is trivial for this method. Along the way, we propose a novel strategy for compiling the controlled-time evolutions needed for spectral sampling by means of Pauli-frame optimizations. We illustrate two physical applications of quantum spectral sampling -- disordered many-body transitions, and mid-spectrum densities of states -- and what postprocessing steps they require beyond the Fourier outputs of the algorithm.
Partial dynamical symmetry (PDS) is an algebraic structure in which a prescribed symmetry is neither exact nor completely broken: a subset of eigenstates keeps good quantum numbers and remains solvable while the rest of the spectrum mixes. PDS is currently identified from spectroscopic data, band-head energies, level systematics, and $B(E2)$ ratios. We ask whether it also has a purely structural signature in the eigenstates, and find that it does, though not in the magnitude of entanglement. The natural diagnostic is the variance of a symmetry Casimir, the label variance $\Var\,\C[G]$, which we show coincides with a block-coherence entropy and a block impurity: all three vanish exactly when a state carries a single irreducible-representation label. Resolved state by state, this quantity is zero on the solvable subset and of order $N^2$ on the mixed states, at stable symmetry points and at Leviatan's first- and second-order critical points, where it takes two distinct forms set by the order of the transition. The magnitude of bipartite entanglement, by contrast, does not separate solvable from mixed states and drifts even where the labels are exact. We anchor the analysis in $^{168}$Er, connect the block purity to the ``purity/coherence''language of the quasi-dynamical-symmetry literature, and show the label variance is uncorrelated with multipartite entanglement and with magic. Finally we encode the model on a qubit register and prepare its solvable and mixed eigenstates variationally, as a step toward evaluating the diagnostic on a quantum device.
We consider a broad family of Heisenberg-type quantum spin systems that were studied by Suzuki and Fisher in 1971. This family includes, for example, the Heisenberg antiferromagnet on any bipartite graph. The ground and Gibbs states of these models are Lee-Yang tensors with radius 1: they are associated with multilinear polynomials that possess an extraordinary zero-freeness property inside the unit polydisk in the complex plane. For each Hamiltonian in this family we show that the spectral gap between the first-excited and ground-state energies is lower bounded by $2\mu$, where $\mu$ is the magnetic field along the $Z$ direction. Using this result we obtain an efficient quantum adiabatic algorithm for the ground energy of any model in this family. The proof is based on a new inequality that relates the spectral gap of a positive semidefinite operator to its Lee-Yang radius -- a quantitative strengthening of prior work of the authors that may find applications elsewhere.
S. Bravyi, David Gosset, Yinchen Liu et al.· 0 citations
We show that the network of quantum register states of a quantum computer (QC), coupled through residual two-body interactions between qubits, exhibits small-world properties analogous to those of complex networks found in human society. The most probable Erd\H{o}s number between any two states is about 9, comparable to the six degrees of separation reported by Milgram for social networks. Using the $\mathring{A}$berg criterion, which retains only interactions exceeding the local energy spacing, we construct an effective ($\mathring{A}$berg) network and show that, above a critical coupling strength, this network percolates into a giant component spanning nearly the whole quantum register space. This percolation transition closely matches the onset of quantum chaos and dynamical thermalization established previously via costly exact diagonalization, while our approach extends the accessible system size up to $n_q=30$ qubits.
The output state of a 2D geometrically local shallow random quantum circuit does not have long range correlations due to its lightcone structure. But this changes if one measures a subset of the qubits: long-range entanglement can be induced by the measurement process, leading to conditional correlations between distant qubits. In this paper we investigate the structure of conditional dependence in these circuits and its consequences for quantum advantage. For a tripartition $ABC$ of the qubits, we consider the ensemble of post-measurement states on $A$ that is conditioned on a specific measurement outcome on $B$ and ranges over all possible measurement outcomes on $C$. For circuit depths exceeding a constant critical value $d^*$, we conjecture that this ensemble is well approximated by a certain generalization of the Haar ensemble, called the Scrooge ensemble~[Jozsa \textit{et al.}, \href{https://doi.org/10.1103/PhysRevA.49.668}{Phys. Rev. A \textbf{49}, 668 (1994)}]; we also provide supporting numerical and analytical evidence. Our conjecture describes a precise sense in which the state retains its lightcone structure on the remaining unmeasured qubits, but also develops some globally random features arising from the measurement. A consequence is that $n$-qubit shallow random quantum circuits in two dimensions are classically efficiently simulable in the presence of a tiny depolarizing noise rate $\Omega(\log(n)/n)$.
Yinchen Liu, Max McGinley, T. Schuster et al.· 1 citation
The quantum capacity of the bosonic thermal attenuator, which is given by the regularization of its coherent information, is unknown. The seminal work of Holevo and Werner established in 1999 the standard one-use lower bound obtained from input thermal states. We first prove that this long-standing lower bound is the exact supremum over all single-mode Gaussian states and then show that, crucially, a non-Gaussian state can do better. As a consequence, we prove positivity of the quantum capacity in a parameter region where the channel is not antidegradable, yet its coherent information optimized over single-mode Gaussian states vanishes. For example, with one thermal photon in the environment and at transmissivity $\eta=0.8$, the coherent information is non-positive for every single-mode Gaussian input. We give an explicit rank-two non-Gaussian state, supported on only six Fock levels, whose coherent information is certified to be at least $4.7\times10^{-4}$ qubits per channel use. This short witness is far from numerically optimal: a numerical optimization over fixed non-Gaussian families reaches at least $8.4\times 10^{-3}$ qubits per channel use at the same point. More generally, at $\nu=1$, using non-Gaussian inputs we certify positivity of the coherent information, and therefore of the quantum capacity, down to $\eta=0.7841$; by contrast, the channel is antidegradable, and hence has zero quantum capacity, for $\eta\leq0.75$. Overall, our work identifies new high-noise regimes in which bosonic quantum communication is possible.
Francesco Anna Mele, Giuseppe Catalano, Marco Fanizza et al.· 0 citations
