Abstract Noise-assisted quantum dynamics can produce useful transport and detection effects, but conclusions drawn from individual stochastic trajectories may not persist under ensemble averaging. We assess a closed-system model in which selected computational-basis states are assigned unit energy in a diagonal Hamiltonian and coupled by a transverse term whose scalar amplitude is independently sampled from a zero-mean Gaussian distribution at each timestep. Four configurations containing 2, 3, and 10 qubits were simulated using sparse exponential propagation. For each original parameter setting, 100 seeded trajectories were compared with zero-perturbation and constant-coupling controls. At the originally proposed weak perturbation strengths, none of the four configurations showed a statistically resolved increase in final marked-state population: all 95% confidence intervals included the uniform baseline. Perturbation-strength sweeps revealed configuration-dependent responses at larger amplitudes, including increased population for one three-qubit configuration, but no universal marked-state amplification or established stochastic-resonance regime. The marked set must also be supplied when constructing the Hamiltonian, so eliminating an explicit phase-oracle circuit does not eliminate its information-encoding role. These results recast the model as an exploratory study of stochastically modulated energy-marked dynamics rather than an oracle-free search algorithm. The work demonstrates the importance of seeded ensembles, controls, and operational detection criteria when evaluating noise-assisted quantum-computing proposals.
Abstract Noise-assisted quantum dynamics can produce useful transport and detection effects, but conclusions drawn from individual stochastic trajectories may not persist under ensemble averaging. We assess a closed-system model in which selected computational-basis states are assigned unit energy in a diagonal Hamiltonian and coupled by a transverse term whose scalar amplitude is independently sampled from a zero-mean Gaussian distribution at each timestep. Four configurations containing 2, 3, and 10 qubits were simulated using sparse exponential propagation. For each original parameter setting, 100 seeded trajectories were compared with zero-perturbation and constant-coupling controls. At the originally proposed weak perturbation strengths, none of the four configurations showed a statistically resolved increase in final marked-state population: all 95% confidence intervals included the uniform baseline. Perturbation-strength sweeps revealed configuration-dependent responses at larger amplitudes, including increased population for one three-qubit configuration, but no universal marked-state amplification or established stochastic-resonance regime. The marked set must also be supplied when constructing the Hamiltonian, so eliminating an explicit phase-oracle circuit does not eliminate its information-encoding role. These results recast the model as an exploratory study of stochastically modulated energy-marked dynamics rather than an oracle-free search algorithm. The work demonstrates the importance of seeded ensembles, controls, and operational detection criteria when evaluating noise-assisted quantum-computing proposals.
Quantum systems out of equilibrium are presently a subject of active research, both in theoretical and experimental domains. In this work, we consider time-periodically modulated quantum systems that are in contact with a stationary environment. Within the framework of a quantum master equation, the asymptotic states of such systems are described by time-periodic density operators. Resolution of these operators constitutes a nontrivial computational task. Approaches based on spectral and iterative methods are restricted to systems with the dimension of the hosting Hilbert space dimH=N≲300, while the direct long-time numerical integration of the master equation becomes increasingly problematic for N≳400, especially when the coupling to the environment is weak. To go beyond this limit, we use the quantum trajectory method, which unravels the master equation for the density operator into a set of stochastic processes for wave functions. The asymptotic density matrix is calculated by performing a statistical sampling over the ensemble of quantum trajectories, preceded by a long transient propagation. We follow the ideology of event-driven programming and construct a new algorithmic realization of the method. The algorithm is computationally efficient, allowing for long "leaps" forward in time. It is also numerically exact, in the sense that, being given the list of uniformly distributed (on the unit interval) random numbers, {η_{1},η_{2},...,η_{n}}, one could propagate a quantum trajectory (with η_{i}'s as norm thresholds) in a numerically exact way. By using a scalable N-particle quantum model, we demonstrate that the algorithm allows us to resolve the asymptotic density operator of the model system with N=2000 states on a regular-size computer cluster, thus reaching the scale on which numerical studies of modulated Hamiltonian systems are currently performed.
We propose a model based on a generalized effective Hamiltonian for studying the effect of noise in quantum computations. The system-environment interactions are taken into account by including stochastic fluctuating terms in the system Hamiltonian. Treating these fluctuations as Gaussian Markov processes with zero mean and $\ensuremath{\delta}$-function correlation times, we derive an exact equation of motion describing the dissipative dynamics for a system of n qubits. We then apply this model to study the effect of noise on the quantum teleportation and a generic quantum controlled-NOT (CNOT) gate. For quantum teleportation, the effect of noise in the quantum channels is found to be additive, and the teleportation fidelity depends on the state of the teleported qubit. The effect of collective decoherence is also studied for the two-qubit entangled states. For the quantum CNOT gate, we study the effect of noise on a set of one- and two-qubit quantum gates, and show that the results can be assembled together to investigate the quality of a quantum CNOT gate operation. We compute the averaged gate fidelity and gate purity for the quantum CNOT gate and investigate phase, bit-flip, and flip-flop errors during the CNOT gate operation. The effects of direct interqubit coupling and fluctuations on the control fields are also studied. We find that the quality of the CNOT gate operation is sensitive to the strengths of the control fields and the strengths of the noise, and the effect of noise is additive regardless of its origin. We discuss the limitations and possible extensions of this model. In sum, we demonstrate a simple model that enables us to investigate the effect of noise in arbitrary quantum circuits under realistic device conditions.
In this thesis, we study the evolution of qubits evolving according to the Schrödinger equation with a Hamiltonian containing noise terms, modeled by random diagonal and off-diagonal matrix elements. For a single qubit exposed to such noise, we show that the noise-averaged qubit density matrix converges to a specific final state, in the limit of large time t. We find that the convergence speed is polynomial in 1=t, with a power that depends on the regularity and the low frequency behaviour of the noise probability density. We evaluate the final state explicitly in the regimes of weak and strong off-diagonal noise. We show that the process implements the well-known dephasing channel in the localized and delocalized basis, respectively. Furthermore, we consider the evolution of the entanglement of two (or more) qubits subject to Gaussian noises with varying means and variances. We consider two different cases: individual noise where each qubit feels an independent noise, and common noise where all qubits are subjected to the same noise. We find the following characteristics of entanglement, measured by the concurrence of qubits. Initially entangled states lose their amount of entanglement in time due to the presence of the noise. The decay of entanglement happens more quickly for common noise than for individual noise. We also detect creation of entanglement due to the common noise: for some initially disentangled states, entanglement is created for intermediate times and then decays to zero in the long time again.
Deterministic Quantum Mechanics (DQM) is a research program investigating whether the probabilistic content of quantum mechanics can be understood as an emergent statistical description of an underlying deterministic physical theory. This paper consolidates the program’s cumulative results into a single account.The central proposal is that a complete microscopic state evolves deterministically, that the quantum state is a coarse-grained ensemble descriptor rather than the complete physical state, and that measurement outcomes are selected by which deterministic basin of an enlarged phase space a microscopic trajectory occupies. Under this proposal, Born probabilities are basin measures rather than fundamental stochastic laws.For a qubit, this program is realized exactly: a hidden variable λ on the unit sphere, a preparation-dependent density μ_r(λ) = [r·λ]₊/π, and the deterministic response A(n,λ) = sgn(n·λ) reproduce the qubit Born rule P(±|r,n) = (1 ± r·n)/2 exactly, together with sequential-measurement statistics and the associated expectation values. This preparation measure is itself shown to be the deterministic pushforward of a uniformly distributed microscopic preparation coordinate, removing any appeal to fundamental preparation-level stochasticity.Measurement is then developed as deterministic basin selection within a finite-time Hamiltonian architecture, with a finite-time stability theorem governing when microscopic perturbations cannot dislodge a trajectory from its basin. Entangled correlations are reproduced by an autonomous, rotationally covariant Bell Hamiltonian containing dynamical setting registers and a conserved-momentum phase shear; this mechanism reproduces the singlet correlation E(a,b) = −a·b and the CHSH value 2√2 in its ideal limit, while preserving exactly uniform local marginals at finite setting inertia and yielding an explicit, quantitatively bounded finite-inertia correction. A finite Hamiltonian bath (Model-C) supplies a deterministic amplification and record-retention mechanism; its analytic structure — including an exact completed-square cancellation of the static bath backreaction and an exact finite-bath spectral equation — is derived independently of, and is shown to survive, a documented numerical implementation failure (the C14 program), which is retained in the record as a diagnosed, non-Hamiltonian right-hand side rather than as evidence against the underlying construction.Extending the program beyond the qubit exposes a genuine mathematical obstruction: no deterministic basin defined directly on the qutrit pure-state manifold ℂP² can reproduce an arbitrary rank-one Born probability, because positivity forces any exact basin onto a measure-zero point. This is proved rigorously. The resolution is not the abandonment of determinism but the enlargement of the hidden state beyond the quantum ray itself, to a genuine mechanical phase space such as the cotangent bundle T*SU(3), on which the quantum state re-emerges as a coarse-grained moment and on which contextual, Kochen–Specker-compatible measurement dynamics can in principle be realized.Throughout, the paper enforces a strict evidentiary hierarchy — mathematical construction, analytic theorem, numerical validation, and experimental confirmation are never conflated — and closes with an explicit account of what has been proved, what remains open, and what would be required to render the program experimentally falsifiable.
We review recent progress in the nonequilibrium dynamics of thermally isolated many-body quantum systems, evolving with an ensemble of Hamiltonians as opposed to deterministic evolution with a single time-dependent Hamiltonian. Such questions arise in (i) quantum dynamics of disordered systems, where different realizations of disorder give rise to an ensemble of real-time quantum evolutions, (ii) quantum evolution with noisy Hamiltonians (temporal disorder), which leads to stochastic Schrödinger equations, and, (iii) in the broader context of quantum optimal control, where one needs to analyze an ensemble of permissible protocols in order to find one that optimizes a given figure of merit. The theme of ensemble quantum evolution appears in several emerging new directions in noneqilibrium quantum dynamics of thermally isolated many-body systems, which include many-body localization, noise-driven systems, and shortcuts to adiabaticity.
The quantum formalism contains a structural discrepancy: the transition amplitude computes the interaction-specific relative phase between probe and target and its effect on coupling strength, while the post-interaction state description does not preserve it. A companion paper identified this discrepancy and argued that the measurement problem arises from mistaking the formalism’s predictive completeness for descriptive completeness at the level of individual events. The present paper develops the quantitative consequence. A minimal phenomenological model shows that, for an event-level conditional law of the stated phase-dependent form, uniform averaging over the interaction-relative phase recovers the Born rule exactly, while partial phase control narrows the outcome distribution according to a visibility-loss factor . Starting from the Jaynes-Cummings Hamiltonian for a two-level system coupled to a single-mode field, we derive the event-level conditional probability , where is the phase-coupling strength — a quantity determined by the microscopic interaction parameters. The phase-coupling strength has two contributions. A pointer-overlap term, , arises from the finite separation of the measurement pointer states and decays exponentially with measurement strength . A persistent term, , arises from beyond-dispersive corrections to the Jaynes-Cummings dynamics. The standard treatment computes the rate of these transitions by squaring the matrix elements, eliminating the relative phase between qubit and field; the present framework retains the amplitude structure prior to ensemble reduction, predicting that the transitions are phase-correlated with the qubit-field relative phase. We propose a specific experiment: sweeping measurement strength while tracking phase-resolved outcome modulation in a circuit QED system. The standard treatment predicts the modulation amplitude decays to zero; the present framework predicts it saturates at a nonzero floor. Because a small phase-dependent bias can also arise from ordinary measurement non-diagonality, the decisive evidence is not the cosine modulation itself but its parameter dependence: the predicted amplitude must scale as across controlled changes in detuning, reverse under , and persist under recalibration. The experiment is feasible with current technology and would determine whether event-level phase structure persists through the measurement interaction or is genuinely absent from the outcome-producing physics.