Deterministic Quantum Jump Method

• The Deterministic Quantum Jump (DQJ) method is a framework that deterministically evolves quantum states between discrete jump events, ensuring consistency with the underlying master equation.
• It replaces stochastic sampling with controlled quadrature and non-Hermitian no‐jump evolution, making it especially useful in weakly dissipative regimes.
• DQJ methods reduce sampling noise and improve efficiency, though they require careful management of combinatorial growth at higher jump orders and memory tradeoffs.

Searching arXiv for the cited DQJ-related papers and nearby terminology. arXiv search: "Deterministic Quantum Jump Method weakly dissipative systems (Meschede et al., 4 Mar 2026)" arXiv search: "Jump probabilities in the non-Markovian quantum jump method (Harkonen, 2010)" The deterministic quantum jump (DQJ) method denotes a class of quantum-trajectory constructions for open quantum systems in which the state evolves deterministically between discrete jump events, while the jump structure is fixed so that the ensemble reproduces the underlying master equation. In the broad trajectory-theory sense, DQJ coincides with a piecewise deterministic process built from a non-Hermitian no-jump evolution and deterministic jump maps. In a narrower recent sense, the term refers to a weak-dissipation algorithm that replaces stochastic sampling of rare jump times by deterministic quadrature over low-jump sectors of the Lindblad evolution. Closely related deterministic formulations also appear in jump-conditioned feedback theory, where one evolves augmented density operators indexed by the last jump record rather than Monte Carlo trajectories [(Harkonen, 2010); (Meschede et al., 4 Mar 2026); (Rosal et al., 2 Jul 2025)].

1. Definition and scope

In the literature, DQJ is not a single formalism but a family of closely related constructions that retain the jump unravelling of open-system dynamics while minimizing or reorganizing the stochastic component. The common structure is that the state update maps are deterministic, whereas randomness enters either through jump times and channels or through the weighting of jump-number sectors. This places DQJ at the intersection of Lindblad dynamics, quantum trajectories, continuous measurement, and feedback control (Meschede et al., 4 Mar 2026, Radaelli et al., 2023).

Context	Deterministic component	Discrete component
Time-local master equations	Non-Hermitian drift of pure states	Forward and reverse jumps
Weakly dissipative Lindblad dynamics	Deterministic quadrature over jump times	$0$-, $1$-, $2$-jump sectors
Continuous-time simulation	Exact waiting-time propagation	Jump time and channel selection
Jump-based feedback	Signal-resolved density-operator evolution	Last jump channel and elapsed time

This variation of usage is substantive rather than terminological. Härkönen’s analysis of time-local non-Markovian master equations gives a general deterministic-plus-jump decomposition of the density operator and derives the jump probabilities used in the non-Markovian quantum jump method (Harkonen, 2010). The 2026 weak-dissipation paper explicitly names the “Deterministic Quantum Jump (DQJ) method” and defines it as a few-jump expansion with deterministic jump-time grids (Meschede et al., 4 Mar 2026). The Gillespie formulation for quantum jump trajectories treats continuous-time exact waiting-time sampling as a DQJ-type method at the density-matrix level, especially beyond purity-preserving unravelings (Radaelli et al., 2023). Deterministic jump equations for feedback go one step further by eliminating sampled trajectories altogether in favor of coupled deterministic equations for signal-resolved states (Rosal et al., 2 Jul 2025).

A useful unifying statement is that DQJ methods preserve the operational meaning of jumps while transferring as much structure as possible into deterministic evolution, deterministic quadrature, or deterministic enlarged-state dynamics.

2. Piecewise-deterministic foundations

A central starting point is the time-local master equation

$$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$

with Hermitian $H(t)$, channel operators $C_k(t)$, and real time-dependent rates $\Delta_k(t)$. Härkönen rewrites the density matrix as an ensemble of pure states on projective Hilbert space,

$$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$

and decomposes the infinitesimal evolution into two processes: deterministic drift of pure states and drift of the probability density functional $P[\psi;t]$ (Harkonen, 2010).

The deterministic part is generated by the effective non-Hermitian Hamiltonian

$$H_\mathrm{eff} = H - \frac{i\hbar}{2}\sum_k \Delta_k C_k^\dagger C_k,$$

followed by renormalization. To first order in $1$0, the state-vector increment is

$1$1

This is the deterministic no-jump evolution used in MCWF and NMQJ, and it is the basic drift component of the DQJ picture (Harkonen, 2010).

The complementary stochastic structure is encoded in the conditional jump probabilities. Splitting each rate as $1$2, with $1$3, Härkönen obtains

$1$4

For $1$5, only forward jumps contribute. For $1$6, reverse jumps appear and become ensemble dependent through $1$7. This is the formal underpinning of the non-Markovian quantum jump method: standard MCWF is recovered in the Markovian limit, whereas negative-rate intervals require reverse-jump rules that encode memory effects and information backflow (Harkonen, 2010).

Two consequences are particularly important for DQJ. First, the dynamics is piecewise deterministic in the precise PDMP sense: deterministic drift between jumps, deterministic state maps at jumps, stochastic timing and channel choice. Second, the unravelling is not unique. Härkönen explicitly notes that a different deterministic propagator $1$8 could be chosen, with compensating changes in the jump process, so the same master equation admits multiple deterministic-jump decompositions (Harkonen, 2010).

3. Weakly dissipative DQJ as a few-jump expansion

The explicit DQJ method of 2026 is designed for the weakly dissipative regime, defined by

$1$9

so that jumps are rare over the time interval $2$0. Instead of sampling many stochastic trajectories and waiting for rare one- and two-jump events to occur, the method decomposes the density matrix by jump number,

$2$1

with $2$2 the probability of exactly $2$3 jumps and $2$4 the normalized conditional state for that sector (Meschede et al., 4 Mar 2026).

The effective Hamiltonian remains the standard

$2$5

and the zero-jump sector is simply the normalized non-Hermitian evolution. The single-jump sector is an integral over jump time $2$6 and jump operator $2$7,

$2$8

with

$2$9

The defining DQJ step is then to replace this integral by a deterministic midpoint quadrature over

$$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$0

and to analogously discretize the two-jump simplex using

$$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$1

where $$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$2 adds barycentric points to better resolve $$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$3 (Meschede et al., 4 Mar 2026).

At first order, the reconstructed density matrix is

$$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$4

with

$$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$5

At second order,

$$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$6

The method therefore removes Monte Carlo fluctuations and replaces them by controlled quadrature error plus truncation error from neglected higher-jump sectors (Meschede et al., 4 Mar 2026).

The asymptotic scaling analysis is one of its distinctive features. For maximal jump order $$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$7,

$$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$8

provided the grid resolves the fastest density-matrix oscillations. Thus the paper reports infidelity scaling $$\frac{ d \rho(t)}{ d t } = -\frac{ i }{ \hbar } [ H(t), \rho(t) ] + \sum_{k} \Delta_k(t) \Big( C_k (t)\rho(t) C_k^\dagger (t) - \frac{1}{2} \big\{ C_k^\dagger (t) C_k (t), \rho (t) \big\} \Big),$$9 for $H(t)$0-jump DQJ and $H(t)$1 for $H(t)$2-jump DQJ, in contrast with stochastic averaging, which yields $H(t)$3 in measures like infidelity (Meschede et al., 4 Mar 2026).

The paper demonstrates this on two benchmark models. For a $H(t)$4-qubit dissipative transverse-field Ising model with $H(t)$5, the $H(t)$6-jump DQJ method shows an asymptotic infidelity scaling close to $H(t)$7, a plateau infidelity of about $H(t)$8, and reaches a given target infidelity with several orders of magnitude fewer trajectories than SQJ (Meschede et al., 4 Mar 2026). For the dissipative Kerr oscillator with $H(t)$9, single-jump DQJ gives significantly smaller time-domain errors than SQJ, and for the integrated spectral quantity $C_k(t)$0 it reaches a plateau about $C_k(t)$1 with as few as a few tens of trajectories, whereas SQJ would require roughly $C_k(t)$2 trajectories to cross the same error threshold suggested by a scaling line (Meschede et al., 4 Mar 2026).

The price of determinism is combinatorial growth with jump order. The number of trajectories is

$C_k(t)$3

at first order and

$C_k(t)$4

at second order, so the method is sharply targeted at low-jump regimes (Meschede et al., 4 Mar 2026).

4. Continuous-time exact simulation and generalized DQJ sampling

A different DQJ line emphasizes exact continuous-time simulation rather than deterministic quadrature. The quantum Gillespie algorithm reformulates the jump unravelling of a Lindblad equation in terms of the waiting-time distribution

$C_k(t)$5

with

$C_k(t)$6

Channel selection at fixed waiting time is then

$C_k(t)$7

This yields exact continuous-time trajectories specified by a sequence of pairs $C_k(t)$8, each followed by deterministic no-jump propagation and deterministic state update (Radaelli et al., 2023).

The main technical device is the operator

$C_k(t)$9

or more generally $\Delta_k(t)$0 for mixed-state dynamics. Precomputing $\Delta_k(t)$1 on a grid allows the waiting-time distribution to be evaluated for arbitrary $\Delta_k(t)$2 through $\Delta_k(t)$3, after which one samples a waiting time in one shot rather than by small-step Bernoulli trials (Radaelli et al., 2023).

This formulation is especially important because it extends DQJ-type simulation beyond purity-preserving unravelings. For partial monitoring or channel merging, the no-jump generator becomes

$\Delta_k(t)$4

and the waiting-time density for the next monitored jump is

$\Delta_k(t)$5

Because these trajectories need not preserve purity, the density matrix itself becomes the trajectory variable, which is not accessible to standard pure-state MCWF in the same way (Radaelli et al., 2023).

The paper positions this as advantageous when the Hilbert space is not huge, a very large number of trajectories is required, and monitored jumps are rare compared with fast coherent dynamics. The tradeoff is explicit: the method trades memory for time by precomputing propagators and adjoint-evolved operators on a time grid (Radaelli et al., 2023). This gives a generalized DQJ picture in which stochasticity is confined to exact event sampling, while the inter-event motion is fully deterministic.

5. Deterministic jump equations for feedback and conditional control

A further development removes sampled trajectories altogether when feedback depends on jump records. In the instrument-based framework of Rosal, Potts, and Landi, one introduces a signal-resolved hybrid state

$\Delta_k(t)$6

which obeys the deterministic recursion

$\Delta_k(t)$7

For quantum jumps, when the signal is the pair $\Delta_k(t)$8 consisting of the last jump channel and the time since the last jump, the continuous-time limit yields

$\Delta_k(t)$9

$$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$0

The unconditional state is recovered as

$$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$1

This is a deterministic quantum-jump equation in the literal sense: jump timing information is encoded as a continuous age variable $$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$2, and channel information is encoded by the discrete label $$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$3, with no Monte Carlo sampling required (Rosal et al., 2 Jul 2025).

When feedback depends only on the last jump channel, the $$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$4-resolved equation reduces to

$$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$5

For the qubit population-inversion example with thermal absorption and emission channels, the formalism yields an analytic optimal drive duration $$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$6, an inversion threshold $$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$7, and a maximum tolerable delay $$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$8 in the low-temperature limit (Rosal et al., 2 Jul 2025).

The experimental counterpart of conditional determinism appears in the superconducting three-level system of Minev and collaborators. There, the onset time of a jump remains stochastic, but once heralded by a lull in the monitored bright-state record, the jump from $$\rho (t) = \int \mathrm{d} \psi\, P[\psi; t] \, |\psi\rangle\langle\psi|,$$9 to $P[\psi;t]$0 follows a continuous, coherent, and deterministic trajectory in the $P[\psi;t]$1–$P[\psi;t]$2 Bloch sphere, and real-time feedback can catch and reverse the jump mid-flight (Minev et al., 2018, Minev, 2019). This experimental result does not define a numerical DQJ algorithm, but it strongly supports the conditional-deterministic interpretation that underlies trajectory-based DQJ methods.

6. Variants, design freedom, and limitations

The DQJ idea is broader than Lindblad few-jump quadrature. In the spatial decoherence master equation,

$P[\psi;t]$3

the orthojump unravelling defines a piecewise deterministic process with nonlinear deterministic evolution

$P[\psi;t]$4

Poisson rate

$P[\psi;t]$5

and jump rule

$P[\psi;t]$6

The jump makes the post-jump state orthogonal to the pre-jump state, hence “orthojump.” Numerical Monte Carlo in units $P[\psi;t]$7 gives asymptotic center-of-mass spreads

$P[\psi;t]$8

showing localization even though jumps continue indefinitely (Homa et al., 2017).

For quadratic Hamiltonians and linear Lindblad operators, quantum jumps can be represented in a Hagedorn-wavepacket basis. The underlying Gaussian parameters evolve deterministically under non-Hermitian propagation, while jumps act algebraically on a coefficient vector in the moving basis. The jump times still come from the usual hazard-rate condition, but the continuous backbone is deterministic and low dimensional. This yields a Gaussian-based DQJ structure without claiming that the jump unravelling itself becomes non-stochastic (Christie et al., 2022).

A different kind of freedom is structural rather than algorithmic. The rate-operator framework of Smirne and collaborators shows that the partition of a master equation into deterministic and jump parts is inherently non-unique. By introducing

$P[\psi;t]$9

one obtains alternative effective Hamiltonians

$$H_\mathrm{eff} = H - \frac{i\hbar}{2}\sum_k \Delta_k C_k^\dagger C_k,$$0

and alternative jump schemes for the same mixed-state dynamics. Under suitable conditions this permits a fixed basis of post-jump states, and in driven systems it can make the deterministic evolution time independent by transferring explicit driving dependence into the jump part. Dissipativity provides a sufficient condition for a representation with positive rate operator $$H_\mathrm{eff} = H - \frac{i\hbar}{2}\sum_k \Delta_k C_k^\dagger C_k,$$1, and hence for a physically meaningful DQJ-style unravelling with a prescribed deterministic generator (Chruściński et al., 2020).

Across these variants, the main limitations recur. The weakly dissipative DQJ method relies on $$H_\mathrm{eff} = H - \frac{i\hbar}{2}\sum_k \Delta_k C_k^\dagger C_k,$$2 and becomes expensive at high jump order because the number of trajectories grows polynomially in $$H_\mathrm{eff} = H - \frac{i\hbar}{2}\sum_k \Delta_k C_k^\dagger C_k,$$3 (Meschede et al., 4 Mar 2026). Continuous-time density-matrix DQJ methods trade sampling noise for precomputation and memory, which can be severe for large Liouvillians (Radaelli et al., 2023). Non-Markovian reverse-jump probabilities are ensemble dependent and therefore operationally more intricate than Markovian MCWF (Harkonen, 2010). More generally, the deterministic/jump split is not unique, so “the” DQJ method is best understood as a design principle for unravelings rather than a single canonical algorithm (Chruściński et al., 2020).

Within that broader principle, DQJ methods provide a rigorous way to exploit the deterministic structure latent in quantum-jump dynamics: deterministic drift between events, deterministic state-update maps at events, deterministic quadrature of rare-event sectors, or deterministic enlarged-state equations for jump-conditioned control.