Random-Unitary Bath Model in Quantum Systems

• The Random-Unitary Bath Model is a framework that represents decoherence (e.g., pure dephasing) as a convex sum of system unitaries, bypassing explicit bath dynamics.
• It maps the system–bath interaction onto a classical stochastic process, allowing efficient simulation and experimental emulation through randomized phase shifts.
• The model underpins practical quantum benchmarking by enabling engineered noise channels and clarifies the boundary between classically simulable and genuinely quantum noise.

The Random-Unitary Bath Model is a construct in open quantum system theory whereby the reduced evolution of a quantum system interacting unitarily with a bath is represented—exactly or approximately—by an ensemble of randomly applied unitaries, rather than explicit bath degrees of freedom. For a broad and technically significant class of noise processes, particularly pure dephasing in qubit or multi-qubit systems, this yields a channel that is mathematically equivalent to a probabilistic (often convex) sum over system unitaries, and thereby admits efficient classical simulation and experimental emulation.

1. Formal Definition and Conceptual Framework

The Random-Unitary Bath Model formalizes the equivalence between two theoretical perspectives on decoherence:

• System–Bath (SB) picture: The system $S$ interacts with an explicit quantum environment $B$ via a joint unitary evolution dictated by a Hamiltonian $H = H_S + H_B + H_{SB}$. The system’s reduced state evolves as

$$\rho_S(t) = \mathrm{Tr}_B \left[ e^{-iHt} \, [\rho_S(0) \otimes \rho_B] \, e^{+iHt} \right].$$

• Random Classical (RC) picture: The bath is replaced by a classical stochastic field $h(t)$ drawn from a probability functional $\mathcal{P}[h]$, generating a random-unitary channel:

$$\rho_S(t) = \int \mathcal{D}[h(t')] \; \mathcal{P}[h(\cdot)] \; U_{\mathrm{cl}}[h(\cdot)] \, \rho_S(0) \, U_{\mathrm{cl}}^\dagger[h(\cdot)].$$

The defining property of the random-unitary bath model is that, for a certain class of Hamiltonians—most notably those generating pure dephasing—the dynamics of the open system are exactly simulable by such a convex mixture over unitary evolutions, without recourse to an explicit environmental Hilbert space (Crow et al., 2013).

2. Constructive Mapping: Single-Qubit Pure Dephasing

Consider the Hamiltonian for single-qubit pure dephasing: $$H_S = -\frac{1}{2}B \sigma_z, \quad H_{SB} = \sigma_z \otimes V_B,$$ where $[H_S, H_{SB}] = 0$, ensuring no population transfer in the computational ($\sigma_z$) basis. The only nontrivial evolution affects the off-diagonal element: $$\rho_{01}(t) = \mathrm{Tr}_{SB}\left[(|0\rangle\langle1| \otimes I_B) \, e^{-iHt} (\rho_S(0) \otimes \rho_B) e^{+iHt} \right].$$ The “noise functional” is defined as

$$F(t) = \mathrm{Tr}_B \left[ e^{-i(H_B + V_B)t} \rho_B \right].$$

The state evolution acquires the form

$$\rho_S(t) = \begin{pmatrix} \rho_{00}(0) & \rho_{01}(0) e^{-iBt} F(t) \ \rho_{10}(0) e^{+iBt} F^*(t) & \rho_{11}(0) \end{pmatrix}.$$

This evolution is equivalent to a two-element random-unitary channel: $$\rho_S(t) = \frac{1}{2} U_1(t) \rho_S(0) U_1^\dagger(t) + \frac{1}{2} U_2(t) \rho_S(0) U_2^\dagger(t)$$ where

$$U_i(t) = \exp\left[-\frac{i}{2} \Phi_i(t) \sigma_z \right],$$

with $\Phi_{1,2}(t)$ determined via the decomposition of the Bloch-plane transfer matrix associated with $F(t)$ (Crow et al., 2013).

3. Explicit Construction and Algorithmic Decomposition

Given the evolution, any real $2\times2$ transfer matrix with $c^2 + s^2 \leq 1$ corresponding to coherence decay may be decomposed as: $T = \frac{1}{2} R(\Phi_1) + \frac{1}{2} R(\Phi_2),$ where $R(\Phi)$ is a rotation and the angle solutions are

$$r = \sqrt{c^2+s^2}, \quad \beta = \frac{\sqrt{1-r^2}}{r},$$

$$\Phi_{1,2} = \tan^{-1}\left( \frac{s\mp\beta c}{c\pm\beta s} \right).$$

The corresponding classical noise fields are $h_{i}(t) = \dot{\Phi}_{i}(t) + B$, each occurring with probability $1/2$. Thus, a pure quantum dephasing channel is exactly realized by stochastically applying one of two unitary phase shifts to the system, governed by these fields.

For quantum channels beyond pure dephasing, an explicit construction based on the reduced density matrix $\rho_S(t)$ permits the identification of a probability distribution over (possibly path-dependent) unitaries that reproduces the full dynamics of the open system. Specifically, the single-qubit channel can always be written as (Halataei, 2017): $$\Phi_t[\rho_S(0)] = \int d\phi\, p(\phi; t) \, U(t; \phi) \, \rho_S(0) \, U(t; \phi)^\dagger,$$ with $p(\phi; t)$ a Gaussian parameterized by a variance related to the coherence decay, and $U(t; \phi)$ constructed to match the reduced dynamics for each trajectory. For amplitude damping and other non-unital noise, the unitaries and the probability law may depend on initial conditions and full state histories.

4. Paradigmatic Models and Experimental Realizations

Analytic and empirical validation of the random-unitary bath model have been demonstrated for:

Model	Hamiltonian/Functionals	Random-Unitary Representation
Spin–Boson Model	$$H=-\tfrac12B\sigma_z+\sigma_z\sum_k(g_k b_k^\dag+g_k^* b_k)+\sum_k \omega_k b_k^\dag b_k$$	Random phases constructed from $F(t)$
Central–Spin Model	$$F(t) = \frac{1}{\sqrt{1+\alpha^2 t^2}} e^{i \arctan(\alpha t)}$$	Two possible classical fields for phase kicks
Quantum–Impurity Model	$F(t) = r(t) e^{i \phi(t)}$	Noise constructed numerically via $\phi(t)$

In experimental settings, arbitrary random-unitary channels for qubits—especially dephasing and amplitude noise—can be engineered by synthesizing time-dependent phase or amplitude fluctuations using in-phase/quadrature modulation of a microwave carrier, as verified in trapped $^{171}$Yb$^+$ ions (Soare et al., 2014). The measured coherence decay under such engineered baths quantitatively agrees with predictions from the filter-function formalism, and the decay rate or visibility envelope is tunable via the spectral properties of the synthesized noise.

5. Extensions: Multi-Qubit and Higher-Dimensional Systems

In multi-qubit systems with dephasing noise arising from a set of commuting operators $\{O_i\}$, if all $O_i$ commute with the total Hamiltonian, the SB dynamics become block-diagonal, and each block undergoes independent phase evolution. The random–unitary bath representation remains valid, with phase factors distributed according to a single real random parameter $\alpha$. The reduced channel for the system is given by

$$\rho_{jk}(t) = \int d\alpha\, p(\alpha)\, e^{-i\alpha \gamma_{jk}(t)}\,\rho_{jk}(0),$$

where $\gamma_{jk}(t) = \theta_j(t) - \theta_k(t)$ encodes relative phase evolution (Crow et al., 2013).

Depolarizing channels on arbitrary-dimensional Hilbert spaces $N$, where $\rho \mapsto (1-p)\rho + p\,I/N$, also admit random-unitary representations by integration over the entire unitary group or, operationally, by summing over a unitary 2-design (such as the Clifford group) (Crow et al., 2013). In such cases, the number of unitaries required for an $\epsilon$-randomizing channel scales only logarithmically in the dimension: $M = O((\log d)/\epsilon^2)$ (Chi et al., 2010).

6. Limitations and Open Problems

The random-unitary representation for open-system dynamics is not universal. It applies exactly when the system-bath interaction is diagonal in a known basis (pure dephasing and unital processes), but not all quantum channels (especially those involving non-unital, non-commuting processes, or amplitude damping) admit a random-unitary structure unless one allows state-dependent or trajectory-wise unitary operators (Halataei, 2017). For single-qubit channels, the construction remains valid for any type of noise, but the required unitaries become explicitly state- and history-dependent.

For higher-dimensional systems, random-unitary representations encounter strict constraints—only unital channels are generally amenable (Halataei, 2017). Extending the model to arbitrary noise channels in higher dimensions typically demands correlated random variables and positivity constraints beyond the scope of convex-sum unitary decompositions.

7. Physical Interpretation and Practical Applications

The random-unitary bath construction implies that decoherence mechanisms due to entanglement with an environment can, in many cases, be simulated entirely by classical randomness: for single-qubit pure dephasing (and certain multi-qubit or depolarizing processes), open-system dynamics have hidden-variable descriptions reducible to random phase kicks or a classical noise source.

Practically, the simulation and emulation of quantum noise via random unitary channels provides a powerful method for benchmarking quantum control protocols and validating quantum hardware, as arbitrary error models may be synthesized programmatically via classical random fields and unitaries (Soare et al., 2014). This also establishes a resource-theoretic separation between “truly quantum” environment-induced maps (not random-unitary) and those that are classically simulable.

Further research aims to delineate more sharply the boundary between random-unitary and genuinely quantum (non-random-unitary) noisy channels, characterize necessary and sufficient conditions for random-unitary representability, and explore minimal bath/resource requirements for arbitrary channel engineering.