Simulating unitary dynamics unavailable in nature by finite perturbative expansion of its Hamiltonian is a straightforward and intuitive approach which often fails when continuous-variable systems are considered. We propose a solution to this problem, which lies in constructing these expansions from suitably bounded operators. We demonstrate this by suggesting an elementary bounded gate based on Rabi coupling and showing its optimality and suitability for approximations of unitary operators.
Simulating unitary dynamics unavailable in nature by finite perturbative expansion of its Hamiltonian is a straightforward and intuitive approach which often fails when continuous-variable systems are considered. We propose a solution to this problem, which lies in constructing these expansions from suitably bounded operators. We demonstrate this by suggesting an elementary bounded gate based on Rabi coupling and showing its optimality and suitability for approximations of unitary operators.
I present a short approach to discretizing continuous unitary evolution. It gives a correspondence in both directions between continuous unitary evolutions and products of unitaries in finite dimensions, and I use it to reduce discrete adiabatic theorems to continuous ones. At fine steps, holding truncated samples of a regulated Hamiltonian on a suitable finite partition gives an approximating product. Conversely, rounding the clock identifies a product of T unit-step samples with a continuous evolution, up to two half-step phases. I bound its operator-norm distance from the original evolution by O(1/T) through cancellation within each time step. This assumes bounded first and second derivatives, bounded spectral width, and eigenvalue differences bounded away from nonzero multiples of 2π, uniformly in T. For a unitary walk with these derivative bounds, a fixed common free spectral arc supplies a Hermitian logarithm path satisfying the comparison hypotheses. Continuous simulation guarantees then transfer to the phase-corrected product, and continuous adiabatic bounds to the product itself, with an explicit additive O(1/T) error and each adiabatic theorem's hypotheses retained. For positive definite quantum linear systems of condition number κ, this comparison and continuous adiabatic theory give a rescaled qubitized walk with O(κ/δ) steps for leakage δ, matching the scaling previously obtained through a discrete adiabatic theorem. These products need no discrete adiabatic theorem.
In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing. The approach proposed in (Marzuoli and Rasetti, 2002, 2005a) relies on the (re)coupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on 'qubits' and Boolean gates. Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of values as well as continuous (angular) variables. Such a framework is an ideal playground to discuss discrete (digital) and analogic computational processes, together with their relationships occurring when a consistent semiclassical limit takes place on discrete quantum gates. When working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum finite states-machines which in turn represent discrete versions of topological quantum computation models. We argue that our model embodies a sort of unifying paradigm for computing inspired by Nature and, even more ambitiously, a universal setting in which suitably encoded quantum symbolic manipulations of combinatorial, topological and algebraic problems might find their 'natural' computational reference model.
Digital-analog is a quantum computational paradigm that employs the natural interaction Hamiltonian of a system as the entangling resource, combined with single qubit gates, to implement universal quantum operations. As in the case of its digital gate-based counterpart, designing digital-analog circuits that employ optimal quantum resources often requires an exceedingly large classical computational time. In this work we find a suboptimal solution to this exponentially large problem, showing that it can be solved within polynomial computational time. In particular, we provide an exact solution for the problem of expressing arbitrary two-body Hamiltonians as the sum of local unitary transformations of an arbitrary Ising Hamiltonian, with the total number of required terms being at most quadratic in system size. This allows us to design a digital-analog simulation protocol that avoids employing numerical optimization over a large parameter space at the preprocessing stage, minimizing computational resources and allowing for further scaling.
Quantum dynamics is usually written relative to a smooth external time parameter, while actual control schedules often contain waits, pulses, gates, resets, and event-like interventions. This paper develops the first unitary part of a sigma-time quantum dynamics framework. We define a bounded-Hamiltonian quantum evolution over a finite-horizon bounded-variation sigma clock with a continuous clock component, flat clock intervals, and a locally finite set of atom times carrying unitary gates. Under these hypotheses the sigma-chronological product exists, is unitary, composes as an evolution family, conserves wave-function norm, is strongly continuous away from atom times, and obeys a sharp atom/gate insertion law. We also prove a fixed-clock perturbation estimate and a bounded isolated-pulse corridor in which shrinking finite-width pulses converge in operator norm to declared atomic gates. The result is deliberately scoped: it does not claim an unbounded Schroedinger-operator theorem, a measurement theorem, an open-system theorem, or a Zeno limit theorem. A finite qubit benchmark with two bounded waits and one atomic gate is included as a reproducibility appendix and verifies unitary defect, trace preservation, Choi positivity, and output-state validity to machine precision
Approximation based on perturbation theory is the foundation for most of the quantitative predictions of quantum mechanics, whether in quantum many-body physics, chemistry, quantum field theory, or other domains. Quantum computing provides an alternative to the perturbation paradigm, yet state-of-the-art quantum processors with tens of noisy qubits are of limited practical utility. Here, we introduce perturbative quantum simulation, which combines the complementary strengths of the two approaches, enabling the solution of large practical quantum problems using limited noisy intermediate-scale quantum hardware. The use of a quantum processor eliminates the need to identify a solvable unperturbed Hamiltonian, while the introduction of perturbative coupling permits the quantum processor to simulate systems larger than the available number of physical qubits. We present an explicit perturbative expansion that mimics the Dyson series expansion and involves only local unitary operations, and show its optimality over other expansions under certain conditions. We numerically benchmark the method for interacting bosons, fermions, and quantum spins in different topologies, and study different physical phenomena, such as information propagation, charge-spin separation, and magnetism, on systems of up to 48 qubits only using an 8+1 qubit quantum hardware. We demonstrate our scheme on the IBM quantum cloud, verifying its noise robustness and illustrating its potential for benchmarking large quantum processors with smaller ones.
Linear combination of Hamiltonian simulation (LCHS) provides an asymptotically optimal framework for the quantum simulation of linear non-unitary dynamics. The LCHS approach approximately implements the Peano--Baker propagator through an integral representation determined by a kernel function. We establish an LCHS approximation theorem for a broadened class of admissible kernel functions, defined by natural complex-analytic conditions. In the standard time-independent block-encoding access model, we explicitly construct, for each target error ε, the unique kernel function in this class minimizing the query-cost functional and show that the minimum of this functional has the asymptotic expansion π2e(logε1−loglogε1+o(1)) as ε↓0. The provably optimal leading coefficient 2e/π improves by a factor of approximately 2.1 on the coefficient obtained empirically by Low and Somma through numerical minimization over their family of optimally scaling kernel functions. We construct a spectrally convergent quadrature with explicit bounds on operator error and linear combination of unitaries (LCU) subnormalization, proving that this asymptotic improvement persists after discretization into a finite linear combination of unitaries suitable for implementation. Our results are based on Hardy space theory and convex analysis.