We give a detailed theoretical derivation of the quantum master equation for the coherent Ising machine. This is a quantum computational network with feedback, that solves NP hard combinatoric problems, including the traveling salesman problem and various extensions and analogs. There are two types of master equation that are obtained, either conditional on the feedback current or unconditional. We show that both types can be accurately simulated in a scalable way using stochastic equations in the positive-P phase-space representation. This depends on the nonlinearity present, and we use parameter values that are typical of current experiments. However, while the two approaches are in excellent agreement, they are not equivalent with regard to efficiency. We find that unconditional simulation has much greater efficiency, and is scalable to larger sizes. This is a case where too much knowledge is a dangerous thing. Conditioning the simulations on the feedback current is not essential to determining the success probability, but it greatly increases the computational complexity. To illustrate the speed improvements obtained with the unconditional approach, we carry out full quantum simulations of the quantum master equation with up to 1000 nodes.
We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed p…
Starting from the Liouville-von Neumann equation, under a weak coupling limit we derive the Lindblad master equation for the one-dimensional quantum Ising model in a Markov approximation and a rotating wave approximation. We also prove that the steady solution of the Lindblad equation is the canonical distribution independent of the dissipation rate.
When part of the environment responsible for decoherence is used to extract information about the decohering system, the preferred {\it pointer states} remain unchanged. This conclusion -- reached for a specific class of models -- is investigated in a general setting of conditional master equations using suitable generalizations of predictability sieve. We also find indications that the einselected states are easiest to infer from the measurements carried out on the environment.
Wave-function Monte Carlo methods are an important tool for simulating quantum systems, but the standard method cannot be used to simulate decoherence in continuously measured systems. Here we present a new Monte Carlo method for such systems. This was used to perform the simulations of a continuously measured nano-resonator in [Phys. Rev. Lett. 102, 057208 (2009)].
We consider the accurate investigation of the energy current and its components, heat and work, in some boundary driven quantum spin systems. The expressions for the currents, as well as the associated Lindblad master equation, are obtained via a repeated interaction scheme. We consider small systems in order to analytically compute the steady distribution to study the current in the steady state. Asymmetrical XXZ and quantum Ising models are detailed analyzed. For the XXZ chain we present cases in which different compositions of heat and work currents, obtained via the repeated interaction pr…
The quantum theory of coherent Ising machines, based on degenerate optical parametric oscillators and measurement-feedback circuits, is developed using the positive $P(α,β)$ representation of the density operator and the master equation. The theory is composed of the c-number stochastic differential equations for describing open dissipative quantum dynamics and the replicator dynamics equations for handling measurement-induced collapse of the density operator. We apply the present theory to simulate two simple Ising spin models and elucidate the unique features of this computing machine.
In this article we present a full description of the quantum Kerr Ising model---a linear optical network of parametrically pumped Kerr non-linearities. We consider the non-dissapative Kerr Ising model and, using variational techniques, show that the energy spectrum is primarily determined by the adjacency matrix in the Ising model and exhibits highly non-classical cat like eigenstates. We then introduce dissipation to give a quantum mechanical treatment of the measurement process based on homodyne detection via the conditional stochastic Schrodinger equation. Finally we identify a quantum adva…
We apply the stochastic master equations (quantum filter) derived by Gough et al. (Proc. 50th IEEE Conference on Decision and Control, 2011) to a system consisting of a cavity driven by a multimode single photon field. In particular, we analyse the conditional dynamics for the problem of cross phase modulation in a doubly resonant cavity. Through the exact integration of the stochastic equations, our results reveal features of the problem unavailable from previous models.
We compare the performance of quantum annealing (QA, through Schrödinger dynamics) and simulated annealing (SA, through a classical master equation) on the $p$-spin infinite range ferromagnetic Ising model, by slowly driving the system across its equilibrium, quantum or classical, phase transition. When the phase transition is second-order ($p=2$, the familiar two-spin Ising interaction) SA shows a remarkable exponential speed-up over QA. For a first-order phase transition ($p \ge 3$, i.e., with multi-spin Ising interactions), in contrast, the classical annealing dynamics appears to remain stu…
We consider a system subject to a quantum optical master equation at finite temperature and study a class of conditional dynamics obtained by monitoring its totally or partially purified environment. More specifically, drawing from the notion that the thermal state of the environment may be regarded as the local state of a lossy and noisy two-mode squeezed state, we consider conditional dynamics ("unravellings") resulting from the homodyne detection of the two modes of such a state. Thus, we identify a class of unravellings parametrised by the loss rate suffered by the environmental two-mode s…
We propose a simplified mathematical construction of the quantum Fourier transform which is suited for systems described by Ising-type Hamiltonians. By contrast to the standard Cooley-Tuckey scheme, which prescribes sequences of CPHASE gates, our implementation is based on one-qubit gates and a free evolution process. We also show how to obtain a quadratic speed-up by applying the conditional interactions simultaneously. Thus rather than O($N^2$) our implementation time scales as O($N$). Finally, we show a realization of our method with homogeneous microwave driven ion traps in a magnetic fiel…
We consider several observers who monitor different parts of the environment of a single quantum system and use their data to deduce its state. We derive a set of conditional stochastic master equations that describe the evolution of the density matrices each observer ascribes to the system under the Markov approximation, and show that this problem can be reduced to the case of a single "super-observer", who has access to all the acquired data. The key problem - consistency of the sets of data acquired by different observers - is then reduced to the probability that a given combination of data…
We introduce a phase-space representation for qubits and spin models. The technique uses an SU(n) coherent state basis, and can equally be used for either static or dynamical simulations. We review previously known definitions and operator identities, and show how these can be used to define an off-diagonal, positive phase-space representation analogous to the positive P-function. As an illustration of the phase-space method, we use the example of the Ising model, which has exact solutions for the finite temperature canonical ensemble in two dimensions. We show how a canonical ensemble for an …
We introduce a quantum generalization of classical kinetic Ising models, described by a certain class of quantum many body master equations. Similarly to kinetic Ising models with detailed balance that are equivalent to certain Hamiltonian systems, our models reduce to a set of Hamiltonian systems determining the dynamics of the elements of the many body density matrix. The ground states of these Hamiltonians are well described by matrix product, or pair entangled projected states. We discuss critical properties of such Hamiltonians, as well as entanglement properties of their low energy state…
Quantum annealing in the transverse-field Ising model (TFIM) with open-system dynamics is known to use thermally-assisted tunneling to drive computation. However, it is still subject to debate whether quantum systems in the presence of decoherence are more useful than those using classical dynamics to drive computation. We contribute to this debate by introducing the perturbed ferromagnetic chain (PFC), a chain of frustrated sub-systems where the degree of frustration scales inversely with the perturbation introduced by a tunable parameter. This gives us an easily embeddable gadget whereby pro…
We derive stochastic master equations for a quantum system interacting with a Bose field prepared in a superposition of continuous-mode coherent states. To determine a conditional evolution of the quantum system we use a collision model with an environment given as an infinite chain of not interacting between themselves qubits prepared initially in a entangled state being a discrete analogue of a superposition of coherent states of the Bose field. The elements of the environment chain interact with the quantum system in turn one by one and they are subsequently measured. We determine a conditi…
We study a four-stroke Otto engine whose working fluid is a quantum Ising chain. The thermodynamic cycle consists in sweeps of the transverse magnetic field occurring in thermal isolation, alternated by thermalisation strokes with reservoirs at different temperatures. The system-environment coupling is modelled in a thermodynamically consistent way by means of a nonlocal Lindblad master equation. We show that the engine may operate in four different operation modes, depending on the various parameters, in particular it can act as a heat engine and as a refrigerator. We detect an enhancement of…
A simple algebraic procedure is described for deriving Maxwell-Bloch-type equations from single-atom cavity quantum electrodynamics (cavity QED) master equations via orthogonal projection onto a manifold of semiclassical states. In particular the usual Maxwell-Bloch Equations are obtained--up to a state-dependent correction factor of order unity--straightforwardly from the unconditional Jaynes-Cummings master equation. The technique of projecting onto a semiclassical manifold can also be applied with conditional master equations (quantum filters), leading to stochastic simulation models that i…
We study the transverse-field Ising model with infinite-range coupling and spontaneous emission on every site. We find that there is spin squeezing in steady state due to the presence of the transverse field. This means that there is still entanglement, despite the decoherence from spontaneous emission. We analytically calculate fluctuations beyond mean-field theory using a phase-space approach, which involves converting the master equation into a Fokker-Planck equation for the Wigner function. Our calculations are relevant to current experiments with trapped ions.
1 First Lecture: Basics 1.1 Physical Derivation of the Master Equation 1.2 Some Simple Implications 1.3 Steady State 1.4 Action to the Left 2 Second Lecture: Eigenvalues and Eigenvectors of L 2.1 A Simple Case First 2.2 The General Case 3 Third Lecture: Completeness of the Damping Bases 3.1 Phase Space Functions 3.2 Completeness of the Eigenvectors of L 3.3 Positivity Conservation 3.4 Lindblad Form of Liouville Operators 4 Fourth Lecture: Quantum-Optical Applications 4.1 Periodically Driven Damped Oscillator 4.2 Conditional and Unconditional Evolution 4.3 Physical Signicance of Statistical Ope…
Many observers can simultaneously measure different parts of an environment of a quantum system in order to find out its state. To study this problem we generalize the formalism of conditional master equations to the multiple observer case. To settle some issues of principle which arise in this context (as the state of the system and of the environment are ultimately correlated), we consider an example of a system qubit interacting through controlled nots (CNOTs) with environmental qubits. The state of the system is the easiest to find out for observers who measure in a basis of the environmen…
We develop from first principles Markovian master equations suited for studying the time evolution of a system evolving adiabatically while coupled weakly to a thermal bath. We derive two sets of equations in the adiabatic limit, one using the rotating wave (secular) approximation that results in a master equation in Lindblad form, the other without the rotating wave approximation but not in Lindblad form. The two equations make markedly different predictions depending on whether or not the Lamb shift is included. Our analysis keeps track of the various time- and energy-scales associated with …
We formulate a multiple-encounter model of the radical pair mechanism that is based on a random coupling of the radical pair to a minimal model environment. These occasional pulse-like couplings correspond to the radical encounters and give rise to both dephasing and recombination. While this is in agreement with the original model of Haberkorn and its extensions that assume additional dephasing, we show how a nonlinear master equation may be constructed to describe the conditional evolution of the radical pairs prior to the detection of their recombination. We propose a nonlinear master equat…
We investigate the critical behavior, both in space and time, of the wetting interface within the coexistence region around the first-order phase transition of a fully-connected quantum Ising model in slab geometry. For that, we employ the Lindblad master equation formalism in which temperature is inherited by the coupling to a dissipative bath, rather than being a functional parameter as in the conventional Cahn's free energy. Lindblad's approach gives not only access to the dissipative dynamics and steady-state configuration of the quantum wetting interface throughout the whole phase diagram…
We theoretically study the conditional counting statistics of electron transport through a system consisting of a single quantum dot (SQD) or coherently coupled double quantum dots (DQD's) monitored by a nearby quantum point contact (QPC) using the generating functional approach with the maximum eigenvalue of the evolution equation matrix method, the quantum trajectory theory method (Monte Carlo method), and an efficient method we develop. The conditional current cumulants that are significantly different from their unconditional counterparts can provide additional information and insight into…
We study the dynamics of entanglement in the quantum Ising chain with dephasing dissipation in a Lindblad master equation form. We consider two unravelings which preserve the Gaussian form of the state, allowing to address large system sizes. The first unraveling gives rise to a quantum-state-diffusion dynamics, while the second one describes a specific form of quantum-jump evolution, suitably constructed to preserve Gaussianity. In the first case we find a crossover from area-law to logarithm-law entanglement scaling and draw the related phase diagram. In the second case we only find logarith…
We show clear evidence of a quadratic speedup of a quantum annealing (QA) Schroedinger dynamics over a Glauber master-equation simulated annealing (SA) for a random Ising model in one dimension, via an equal-footing exact deterministic dynamics of the Jordan-Wigner fermionized problems. This is remarkable, in view of the arguments of Katzgraber et al., PRX 4, 021008 (2014), since SA does not encounter any phase transition, while QA does. We also find a second remarkable result: that a "quantum-inspired" imaginary-time Schroedinger QA provides a further exponential speedup, i.e., an asymptotic …
We analyze and compare three different schemes that can be used to generate entanglement between spin qubits in optically-active single solid-state quantum systems. Each scheme is based on first generating entanglement between the spin degree of freedom and either the photon number, the time bin, or the polarization degree of freedom of photons emitted by the systems. We compute the time evolution of the entanglement generation process by decomposing the dynamics of a Markovian master equation into a set of propagation superoperators conditioned on the cumulative detector photon count. We then…
In this paper we approach the theory of continuous measurements and the associated unconditional and conditional (stochastic) master equations from the perspective of quantum information and quantum computing. We do so by showing how the continuous-time evolution of these master equations arises from discretizing in time the interaction between a system and a probe field and by formulating quantum-circuit diagrams for the discretized evolution. We then reformulate this interaction by replacing the probe field with a bath of qubits, one for each discretized time segment, reproducing all of the …
In order to be able to study dissipation, the interaction between a single system and their environment was introduced in quantum mechanics. Master and quantum Langeving equations was derived and, also, decoherence was studied using this approach. One of the most used model in this field is a single harmonic oscillator interacting with a reservoir. In this work we solve analytically this problem in the resonance case with the evolution operator method. We use this result to study the conditional dynamics of a finite system of coupling, a bandgap quantum coupler. We study the conditional dynami…
Deep quantum neural networks may provide a promising way to achieve quantum learning advantage with noisy intermediate scale quantum devices. Here, we use deep quantum feedforward neural networks capable of universal quantum computation to represent the mixed states for open quantum many-body systems and introduce a variational method with quantum derivatives to solve the master equation for dynamics and stationary states. Owning to the special structure of the quantum networks, this approach enjoys a number of notable features, including the absence of barren plateaus, efficient quantum analo…
We apply a recently developed theory for metastability in open quantum systems to a one-dimensional dissipative quantum Ising model. Earlier results suggest this model features either a non-equilibrium phase transition or a smooth but sharp crossover, where the stationary state changes from paramagnetic to ferromagnetic, accompanied by strongly intermittent emission dynamics characteristic of first-order coexistence between dynamical phases. We show that for a range of parameters close to this transition/crossover point the dynamics of the finite system displays pronounced metastability, i.e.,…
We investigate a novel type of conditional dynamic that occurs in the strongly-driven Jaynes-Cummings model with dissipation. Extending the work of Alsing and Carmichael [Quantum Opt. {\bf 3}, 13 (1991)], we present a combined numerical and analytic study of the Stochastic Master Equation that describes the system's conditional evolution when the cavity output is continuously observed via homodyne detection, but atomic spontaneous emission is not monitored at all. We find that quantum jumps of the atomic state are induced by its dynamical coupling to the optical field, in order retroactively t…
Dynamics of large complex systems, such as relaxation towards equilibrium in classical statistical mechanics, often obeys a master equation. The equation significantly simplifies the complexities but describes essential information of occupation probabilities. A related fundamental question is the thermalization, a coherent evolution of an isolated many-body quantum state into a state that seems to be in thermal equilibrium. It is valuable to find an effective equation describing this complex dynamics. Here, we experimentally investigate the question by observing sudden quench dynamics of quan…
This article focuses on the general theory of open quantum systems in the Gaussian regime and explores a number of diverse ramifications and consequences of the theory. We shall first introduce the Gaussian framework in its full generality, including a classification of Gaussian (also known as "general-dyne") quantum measurements. In doing so, we will give a compact proof for the parametrisation of the most general Gaussian completely positive map, which we believe to be missing in the existing literature. We will then move on to consider the linear coupling with a white noise bath, and derive…
In this work we first derive a generalized conditional master equation for quantum measurement by a mesoscopic detector, then study the readout characteristics of qubit measurement where a number of new features are found. The work would in particular highlight the qubit spontaneous relaxation effect induced by the measurement itself rather than an external thermal bath.
We present quantitative predictions for quantum simulator experiments on Ising models from trapped ions to Rydberg chains and show how the thermalization, and thus decoherence times, can be controlled by considering common, independent, and end-cap couplings to the bath. We find (i) independent baths enable more rapid thermalization in comparison to a common one; (ii) the thermalization timescale depends strongly on the position in the Ising phase diagram; (iii) for a common bath larger system sizes show a significant slow down in the thermalization process; and (iv) finite-size scaling indica…
We derive the quantum trajectory or stochastic (conditional) master equation for a single superconducting Cooper-pair box (SCB) charge qubit measured by a single-electron transistor (SET) detector. This stochastic master equation describes the random evolution of the measured SCB qubit density matrix which both conditions and is conditioned on a particular realization of the measured electron tunneling events through the SET junctions. Hence it can be regarded as a Monte Carlo method that allows us to simulate the continuous quantum measurement process. We show that the master equation for the…
We present a theoretical study of a superconducting charge qubit dispersively coupled to a transmission line resonator. Starting from a master equation description of this coupled system and using a polaron transformation, we obtain an exact effective master equation for the qubit. We then use quantum trajectory theory to investigate the measurement of the qubit by continuous homodyne measurement of the resonator out-field. Using the same porlaron transformation, a stochastic master equation for the conditional state of the qubit is obtained. From this result, various definitions of the measur…
We obtain the finite-temperature unconditional master equation of the density matrix for two coupled quantum dots (CQD) when one dot is subjected to a measurement of its electron occupation number using a point contact (PC). To determine how the CQD system state depends on the actual current through the PC device, we use the so-called quantum trajectory method to derive the zero-temperature conditional master equation. We first treat the electron tunneling through the PC barrier as a classical stochastic point process (a quantum-jump model). Then we show explicitly that our results can be exte…
One of the remarkable notions in the recent development of quantum physics is the weak value related to weak measurements. We emulate it as a two-time conditional expectation in a classical stochastic model. We use the well known symmetrized form of the master equation, which is formally equivalent to the wave equation in quantum mechanics apart from the fact that wave functions are always real. The origin of the unusual behaviors of the weak value such as the negative probability and the abnormal enhancement of some expectations becomes clearer in the present case, where the two-time conditio…
We present an approach to the numerical simulation of open quantum many-body systems based on the semiclassical framework of the discrete truncated Wigner approximation. We establish a quantum jump formalism to integrate the quantum master equation describing the dynamics of the system, which we find to be exact in both the noninteracting limit and the limit where the system is described by classical rate equations. We apply our method to simulation of the paradigmatic dissipative Ising model, where we are able to capture the critical fluctuations of the system beyond the level of mean-field t…
