Parallel-in-time quantum simulation via Page and Wootters quantum time

TL;DR

This work reframes time as a quantum degree of freedom via the Page–Wootters history-state formalism and develops quantum-algorithmic schemes for parallel-in-time simulations. By encoding temporal indices in a clock-qubit register, it enables the computation of infinite-time temporal averages and dynamical correlators with an exponential trade-off between time and space, controlled by the number of clock qubits $\log(N)$ and the system size. A key insight is that system–time entanglement (the history state's bipartite correlations) encodes meaningful dynamical information and bounds equilibration-related quantities, including the Loschmidt echo and temporal fluctuations of observables. The paper provides both direct and variational-depth strategies, including a Cartan-decomposition approach that achieves $\mathcal{O}(\log(N)n)$ circuit depth for history-state preparation, and numerical demonstrations on Aubry–André and many-body localization models. These results open avenues for studying dynamical properties of many-body quantum systems with reduced circuit depth and offer new perspectives on equilibration, non-equilibrium dynamics, and the role of time in quantum computation.

Abstract

In the past few decades, researchers have created a veritable zoo of quantum algorithms by drawing inspiration from classical computing, information theory, and even from physical phenomena. Here we present quantum algorithms for parallel-in-time simulations that are inspired by the Page and Wootters formalism. In this framework, and thus in our algorithms, the classical time-variable of quantum mechanics is promoted to the quantum realm by introducing a Hilbert space of ``clock'' qubits which are then entangled with the ``system'' qubits. We show that our algorithms can compute temporal properties over $N$ different times of many-body systems by only using $\log(N)$ clock qubits. As such, we achieve an exponential trade-off between time and spatial complexities. In addition, we rigorously prove that the entanglement created between the system qubits and the clock qubits has operational meaning, as it encodes valuable information about the system's dynamics. We also provide a circuit depth estimation of all the protocols, showing a running time advantage in computation times over traditional sequential-in-time algorithms. In particular, for the case when the dynamics are determined by the Aubry--Andre model, we present a hybrid method for which our algorithms have a depth that only scales as $\mathcal{O}(\log(N)n)$. As a by-product, we can relate the previous schemes to the problem of equilibration of an isolated quantum system, thus indicating that our framework enables a new dimension for studying dynamical properties of many-body systems.

Figures (22)

• Figure 1: Quantum algorithms based on the standard quantum mechanics or the PaW formalism. a) In Hamiltonian classical mechanics, dynamical variables are functions of the phase space coordinates position $\boldsymbol{x}$ and momentum $\boldsymbol{p}$. In standard quantum mechanics, one promotes $\boldsymbol{x}$ and $\boldsymbol{p}$ to quantum operators, but the time variable $t$ is treated as a classical parameter that is external to the quantum system being studied. Quantum algorithms for studying dynamical properties based on this framework are implemented for some fixed time $t$. If we want to compute an average of $N$ times, we need to repeat the run $N$ different sequential-in-time experiments. b) In the PaW formalism, time is treated as a quantum variable, with its own associated Hilbert space. In this work we present quantum algorithms for parallel-in-time simulations which trade circuit repetitions for ancilla clock qubits. c) After a proper entangling protocol one has access not only to properties of the system at a given time, but also to their complete history. This information can be retrieved by performing measurements at the end of the circuit, which now can involve either the clock qubits, the system qubits or both. Measurements on the system which are conditioned to a certain time value give properties of the system at a given time. More interestingly, if one only measures on the system and complete ignore the clock's values, temporal averages are obtained. This is a consequence of the entanglement between the system and the clock which induces a useful quantum channel when the clock is treated as an environment. Because of the quantum nature of the simulated clock and system, many other measurements can be proposed, meaning that the different protocols we discuss in this manuscript do not exhaust all the possibilities opened by this computational framework.
• Figure 1: Algorithm for computing the overlap between two quantum states. We show an algorithm which takes as input two arbitrary $n$-qubit quantum states $\rho$ and $\sigma$ and which estimates the overlap $\Tr[\rho\sigma]$. In a) we show the algorithm for the case when $\rho$ and $\sigma$ and single qubit states, and in b) the generalization for larger qubit sizes. We can see that in all cases the circuit depth is equal to two, and hence independent of $n$.
• Figure 2: Circuit for preparing history states. As shown above, the initial state to the clock qubits is $\ket{0}^{\otimes \log(N)}$ while that of the system is $\ket{\psi_{0}}$. The action of the Hadamard gates is to map the initial state to $\ket{+}^{\otimes \log(N)}\otimes \ket{\psi_{0}}$. Here, we find it convenient to write $\ket{+}^{\otimes \log(N)}=\frac{1}{\sqrt{N}}\otimes_{j=1}^{\log{N}}(\ket{0_j}+\ket{1_j})=\frac{1}{\sqrt{N}}\sum_{t=0}^{N-1}\ket{t}$ where we have expressed $t$ in its binary form $t=\sum_{j=1}^{\log{N}}t_j2^{j-1}$. Next, the $\log{N}$ controlled gates $U(2^{j-1}\frac{T}{N})=U(\frac{T}{N})^{2^{j-1}}$ for $j=1,\ldots,\log{N}$ perform the operations $U(\varepsilon t)\ket{\psi_0}=\ket{\psi(\varepsilon t)}$ for $U(\varepsilon t)=U(\frac{T}{N})^{\sum_{j=1}^{\log{N}}t_j2^{j-1}}$.
• Figure 2: Scaling of Bond Dimension with time. We report the scaling with time of the bond dimension $\chi$ of the MPS used to represent the interacting model state, with $\Delta=0.05$. The black, dashed line shows the non-interacting model ($\Delta=0$) bond dimension, which does not depend on $\lambda$ and is equal to $\chi=4$ throughout the whole system's evolution. Instead, in the interacting case, the maximum value of $\chi$ becomes smaller as the localization strength $\lambda$ increases, confirming that the stronger the localization the easier the description of the system.
• Figure 3: Trade-off between accuracy and resolution. Consider the approximation of the infinite-time average of Eq. \ref{['eq:formula-F']} given by the discrete sum in Eq. \ref{['eq:discrete-time']}. For some fix number of time-steps $N$, there exists a trade-off between the window size $\varepsilon$ and the final time $T$. Namely, larger $T$ implies larger window size $\varepsilon$, and hence less accuracy. On the other hand, smaller final time $T$ implies more resolution in the temporal average at the cost of less accuracy.

Theorems & Definitions (22)

• Proposition 1
• Theorem 1
• Proposition 2
• Theorem 2
• Theorem 3
• Corollary 1
• Corollary 2
• Corollary 3
• Theorem 4
• Theorem 5