Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state $\mid \psi ⟩$ of the quantum system. Such expectation values can be measured by repeatedly preparing $\mid \psi ⟩$ and coupling the system to an apparatus. For this method, the precision of the measured value scales as $\frac{1}{\sqrt{N}}$ for $N$ repetitions of the experiment. For the problem of estimating the parameter $\varphi$ in an evolution $e^{-i\varphi H}$, it is possible to achieve precision $\frac{1}{N}$ [the quantum metrology limit; see Giovannetti et al., Phys. Rev. Lett. 96, 010401 (2006)] provided that sufficient information about $H$ and its spectrum is available. We consider the more general problem of estimating expectations of operators $A$ with minimal prior knowledge of $A$. We give explicit algorithms that approach precision $\frac{1}{N}$ given a bound on the eigenvalues of $A$ or on their tail distribution. These algorithms are particularly useful for simulating quantum systems on quantum computers because they enable efficient measurement of observables and correlation functions. Our algorithms are based on a method for efficiently measuring the complex overlap of $\mid \psi ⟩$ and $U\mid \psi ⟩$, where $U$ is an implementable unitary operator. We explicitly consider the issue of confidence levels in measuring observables and overlaps and show that, as expected, confidence levels can be improved exponentially with linear overhead. We further show that the algorithms given here can typically be parallelized with minimal increase in resource usage.

We present a systematic recipe for generating and classifying duality transformations in one-dimensional quantum lattice systems. Our construction emphasizes the role of global symmetries, including those described by Abelian and non-Abelian groups but also more general categorical symmetries. These symmetries can be realized as matrix product operators that allow the extraction of a fusion category that characterizes the algebra of all symmetric operators commuting with the symmetry. Known as the bond algebra, its explicit realizations are classified by module categories over the fusion category. A duality is then defined by a pair of distinct module categories giving rise to dual realizations of the bond algebra, as well as dual Hamiltonians. Symmetries of dual models are, in general, distinct but satisfy a categorical Morita equivalence. A key novelty of our categorical approach is the explicit construction of matrix product operators that intertwine dual bond algebra realizations at the level of the Hilbert space and, in general, map local order operators to nonlocal string-order operators. We illustrate this approach for known dualities such as the Kramers-Wannier, Jordan-Wigner, and Kennedy-Tasaki dualities and the interaction-round-the-face–vertex correspondence, a new duality of the $t$-$J_z$ chain model, and dualities in models with the exotic Haagerup symmetry. Finally, we comment on generalizations to higher dimensions.

We introduce a perturbative approach to solving the time-dependent Schrödinger equation, named adiabatic perturbation theory (APT), whose zeroth-order term is the quantum adiabatic approximation. The small parameter in the power series expansion of the time-dependent wave function is the inverse of the time it takes to drive the system’s Hamiltonian from the initial to its final form. We review other standard perturbative and nonperturbative ways of going beyond the adiabatic approximation, extending and finding exact relations among them, and also compare the efficiency of those methods against the APT. Most importantly, we determine APT corrections to the Berry phase by use of the Aharonov-Anandan geometric phase. We then solve several time-dependent problems, allowing us to illustrate that the APT is the only perturbative method that gives the right corrections to the adiabatic approximation. Finally, we propose an experiment to measure the APT corrections to the Berry phase and show, for a particular spin-$1∕2$ problem, that to first order in APT the geometric phase should be two and a half times the (adiabatic) Berry phase.

What distinguishes trivial superfluids from topological superfluids in interacting many-body systems where the number of particles is conserved? Building on a class of integrable pairing Hamiltonians, we present a number-conserving, interacting variation of the Kitaev model, the Richardson-Gaudin-Kitaev chain, that remains exactly solvable for periodic and antiperiodic boundary conditions. Our model allows us to identify fermion parity switches that distinctively characterize topological superconductivity (fermion superfluidity) in generic interacting many-body systems. Although the Majorana zero modes in this model have only a power-law confinement, we may still define many-body Majorana operators by tuning the flux to a fermion parity switch. We derive a closed-form expression for an interacting topological invariant and show that the transition away from the topological phase is of third order.

Much evidence seems to suggest the cortex operates near a critical point, yet a single set of exponents defining its universality class has not been found. In fact, when critical exponents are estimated from data, they widely differ across species, individuals of the same species, and even over time, or depending on stimulus. Interestingly, these exponents still approximately hold to a dynamical scaling relation. Here we show that the theory of quasicriticality, an organizing principle for brain dynamics, can account for this paradoxical situation. As external stimuli drive the cortex, quasicriticality predicts a departure from criticality along a Widom line with exponents that decrease in absolute value, while still holding approximately to a dynamical scaling relation. We use simulations and experimental data to confirm these predictions and describe new ones that could be tested soon.

A subtle relation between quantum Hall physics and the phenomenon of pairing is unveiled. By use of second quantization, we establish a connection between (i) a broad class of rotationally symmetric two-body interactions within the lowest Landau level and (ii) integrable hyperbolic Richardson-Gaudin-type Hamiltonians that arise in $({p}_{x}+i{p}_{y})$ superconductivity. Specifically, we show that general Haldane pseudopotentials (and their sums) can be expressed as a sum of repulsive noncommuting $({p}_{x}+i{p}_{y})$-type pairing Hamiltonians. The determination of the spectrum and individual null spaces of each of these noncommuting Richardson-Gaudin-type Hamiltonians is nontrivial yet is Bethe ansatz solvable. For the Laughlin sequence, it is observed that this problem is frustration free and zero-energy ground states lie in the common null space of all of these noncommuting Hamiltonians. This property allows for the use of a new truncated basis of pairing configurations in which to express Laughlin states at general filling factors. We prove separability of arbitrary Haldane pseudopotentials, providing explicit expressions for their second quantized forms, and further show by explicit construction how to exploit the topological equivalence between different geometries (disk, cylinder, and sphere) sharing the same topological genus number, in the second quantized formalism, through similarity transformations. As an application of the second quantized approach, we establish a ``squeezing principle'' that applies to the zero modes of a general class of Hamiltonians, which includes but is not limited to Haldane pseudopotentials. We also show how one may establish (bounds on) ``incompressible filling factors'' for those Hamiltonians. By invoking properties of symmetric polynomials, we provide explicit second quantized quasihole generators; the generators that we find directly relate to bosonic chiral edge modes and further make aspects of dimensional reduction in the quantum Hall systems precise.