Quantum Noise Simulation

A numerical noise model specifies correlations, rates, and how noise couples to a system. A label such as “dephasing noise” is insufficient: white, slowly varying, and finite-correlation-time noise can produce different decay laws. This page concerns time-domain physical noise; circuit noise models addresses gate placement and device abstractions.

Helpful background. Lindblad solvers treats Markovian evolution; stochastic integration explains Wiener scaling.

Relaxation and transverse decay

For a qubit in basis $(|0\rangle,|1\rangle)$ use $L_1=\sqrt{\gamma_1}|0\rangle\langle1|$ and $L_\phi=\sqrt{\gamma_\phi/2}\,Z$. Then

$$T_1^{-1}=\gamma_1,\qquad T_2^{-1}=\gamma_1/2+\gamma_\phi.$$

The factor $1/2$ inside $L_\phi$ makes off-diagonal entries decay at rate $\gamma_\phi$. A convention with $L_\phi=\sqrt{\gamma_\phi}Z$ instead doubles that rate. State the operator, not only a symbol called “dephasing rate.”

In this zero-temperature Markovian model $T_2\leq2T_1$. Measured parameters violating this relation cannot be inserted into this model by choosing a negative pure-dephasing rate. They may reflect inconsistent fits, different conditions, or a model outside these assumptions.

Classical frequency noise

Let

$$H(t)=\frac{\hbar}{2}[\omega_0+\xi(t)]Z, \qquad \phi(t)=\int_0^t\xi(s)\,ds.$$

For a zero-mean Gaussian process, averaging the random phase gives a coherence factor

$$W(t)=\mathbb E[e^{-i\phi(t)}] =\exp[-\tfrac12{\rm Var}\,\phi(t)].$$

This follows from the Gaussian characteristic function. It is exact here because Hamiltonians at different times commute; it is not a general replacement for time ordering in driven systems.

For stationary Ornstein–Uhlenbeck noise with $\mathbb E[\xi(t)\xi(0)]=\sigma^2e^{-|t|/\tau_c}$,

$$W(t)=\exp\left\{-\sigma^2 \left[\tau_c t-\tau_c^2(1-e^{-t/\tau_c})\right]\right\}.$$

At $t\ll\tau_c$ decay is Gaussian, $e^{-\sigma^2t^2/2}$; at long times its leading exponent is $-\sigma^2\tau_c t$. This tests both the generated correlation time and the propagator.

Sampling colored noise

Initialize $\xi_0$ from $\mathcal N(0,\sigma^2)$ for a stationary process. The exact grid-point update is

$$\xi_{n+1}=a\xi_n+\sigma\sqrt{1-a^2}\,\eta_n, \qquad a=e^{-\Delta t/\tau_c},$$

with independent unit Gaussians $\eta_n$. Starting at zero introduces a transient covariance and changes the intended ensemble.

Although this update samples grid-point values exactly, approximating $\int\xi\,dt$ by a rectangle or trapezoid rule still introduces an error. Refine the noise sampling interval as well as the quantum integration interval. For noncommuting drives the fastest Hamiltonian and noise scales both matter.

For the commuting qubit benchmark, one can avoid this quadrature error entirely. Write $h=\Delta t$, $\tau=\tau_c$, and sample the endpoint and bin integral together:

$$\xi_{n+1}=a\xi_n+\zeta_n,\qquad I_n=\int_{t_n}^{t_n+h}\xi(s)\,ds =\tau(1-a)\xi_n+\chi_n.$$

The pair $(\zeta_n,\chi_n)$ is a zero-mean Gaussian independent of the past, with covariance

$$\begin{aligned} \mathbb E[\zeta_n^2]&=\sigma^2(1-a^2),\\ \mathbb E[\zeta_n\chi_n]&=\sigma^2\tau(1-a)^2,\\ \mathbb E[\chi_n^2]&=\sigma^2 \left[2\tau h-\tau^2(3-4a+a^2)\right]. \end{aligned}$$

These expressions follow by integrating $d\xi=-\xi\,dt/\tau+\sqrt{2\sigma^2/\tau}\,dW$: the endpoint noise has kernel $e^{-(h-s)/\tau}$, while the integral noise has kernel $\tau[1-e^{-(h-s)/\tau}]$. Integrating their products gives the three covariances. Draw two independent standard normals and multiply by a factor of this $2\times2$ covariance matrix. Sampling $\zeta_n$ and $\chi_n$ independently would destroy correlations across successive phase increments.

Accumulate $\phi_{n+1}=\phi_n+I_n$ to obtain exact phase samples at the grid times for this model (Gillespie, 1996). At very small $h/\tau$, evaluate exponent differences with expm1 and use a series where necessary; in particular $\mathbb E[\chi_n^2]=2\sigma^2h^3/(3\tau)+O(h^4/\tau^2)$. Direct subtraction of nearly equal terms can produce a spurious negative variance. The joint update does not remove quantum propagation error when another Hamiltonian term fails to commute with $Z$.

White noise is a distributional limit. If $\langle\xi(t)\xi(s)\rangle=2D\delta(t-s)$, phase increments have variance $2D\Delta t$, not variance proportional to $\Delta t^2$. Sampling frequency noise with a fixed variance at ever smaller steps would suppress its integrated effect.

Validation and interpretation

Check the empirical noise autocovariance before propagating states. Then compare the undriven qubit against the analytic $W(t)$, using independent realizations for confidence intervals. Finally add driving and refine again. Increasing realization count removes sampling error but not an incorrect covariance or time discretization.

Classical random Hamiltonians produce averages of unitaries and therefore unital channels. They cannot by themselves reproduce zero-temperature amplitude damping to a unique pure ground state. Add a physically justified dissipative mechanism when energy relaxation is required.

Exercise

For quasistatic Gaussian noise, take $\xi(t)=\xi_0$ throughout each run with variance $\sigma^2$. Find the coherence at time $t$.

Solution

The phase has variance $\sigma^2t^2$, so $W(t)=e^{-\sigma^2t^2/2}$. Resampling $\xi_0$ at every integration step instead changes the model. This provides a direct test that a simulator preserves the requested inter-time correlation.

References

• Gillespie, D. T., “Exact numerical simulation of the Ornstein–Uhlenbeck process and its integral,” Physical Review E 54, 2084–2091 (1996), doi:10.1103/PhysRevE.54.2084.
• Jacobs, K., and Steck, D. A., “A straightforward introduction to continuous quantum measurement,” Contemporary Physics 47, 279–303 (2006), arXiv:quant-ph/0611067 — stochastic conventions and measurement noise.
• Uhlenbeck, G. E., and Ornstein, L. S., “On the theory of the Brownian motion,” Physical Review 36, 823–841 (1930), doi:10.1103/PhysRev.36.823 — finite-correlation-time Gaussian process.