Deterministic quantum trajectory via imaginary time evolution

Abstract

Stochastic quantum trajectories, such as pure state evolutions under unitary dynamics and random measurements, offer a crucial ensemble description of many-body open system dynamics. Recent studies have highlighted that individual quantum trajectories also encode essential physical information. Prominent examples include measurement induced phase transitions, where a pure quantum state corresponding to fixed measurement outcomes (trajectories) exhibits distinct entanglement phases, depending on the measurement rate. However, direct observation of this effect is hindered by an exponential post-selection barrier, whereby the probability of realizing a specific trajectory is exponentially small. We propose a deterministic method to efficiently prepare quantum trajectories in polynomial time using imaginary time evolution and, thus, overcome this fundamental challenge. We demonstrate that our method applies to a certain class of quantum states, and argue that there does not exist universal approaches for any quantum trajectories. Our result paves the way for experimentally exploring the physics of individual quantum trajectories at scale and enables direct observation of certain post-selection-dependent phenomena.

Figures (4)

• Figure 1: Left: a random quantum circuit is subject to single-qubit measurements (black dots) at a rate $p$. Depending on the value of $p$, the final state of the qubits can exhibit different phases with distinct entanglement patterns. For $p$ greater than the critical value $p_c$, the state of the qubits exhibits area law entanglement, i.e., the subsystem von Neumann entropy is proportional to the area of the boundary of the subsystem. Whereas for $p<p_c$, the entanglement satisfies the volume law. Right: we consider a one-dimensional qubit system in a periodic boundary condition. The cluster correlation can be quantified by the mutual information between a local subsystem $A$ and $C$, denoted by $I(A,C)(r)$, where $C$ is the cluster that includes all qubits whose distance to $A$ is greater than $r$.
• Figure 2: Decay of $I(A,C)(r)$ in the output state of $n=64$ qubit circuit with $L=n$ layers and at various measurement rates $p$. The critical measurement rate is $p_c = 0.16$Li2019Measurement.
• Figure 3: Cluster correlations at fixed $r=n/16$ for various values of measurement rates $p$ and number of qubits $n$. The data collapse parameter $\nu$ is obtained by minimizing the fitting error. For the area law entanglement regime, $\nu = 1.70$ with an exponential fitting (solid curve) $f(x)=2.68\exp{(-0.42x^{0.68})}$. For the volume law entanglement regime, $\nu = 0.75$ with an exponential fitting (dashed curve) $f(x)=1.88\exp{(-0.04x^{1.06}) + 0.21}$. The critical rate is $p_c=0.16$.
• Figure 4: The infidelity $1 - F(\ket{\psi_{\beta}}, \ket{\psi_{\infty}})$ versus $\beta$ for various $r$. Here, $\ket{\psi_{\beta}}$ is the output of DQITE on $\ket{\psi}$ that is the output of a $n=20$ qubit measured random quantum circuits with $L=n$ layers with measurement rates $p=0.5$ (solid markers) and $p=0.1$ (empty markers). $\ket{\psi_{\infty}}$ is the post-measurement state upon measuring the first qubit in $\ket{\psi}$. For the $p=0.5$ case, the expected infidelity between $\ket{\psi_{\infty}}$ and the exactly (nonunitary) imaginary-time-evolved state to finite $\beta$ is shown as the black curve, confirming the exponential scaling in \ref{['eq:fidelity']}.

Theorems & Definitions (2)

• Proposition 1
• proof