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 that 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 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.
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 that 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 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.
Simulating quantum many-body systems is essential for understanding complex phenomena in condensed matter, chemistry, and materials science. Quantum computing offers a way to overcome the limitations of classical methods by operating directly in exponentially large Hilbert spaces. In particular, classical methods often fail to capture the dynamics of strongly correlated systems, which makes quantum algorithms a promising alternative for exploring these challenging regimes. This thesis uses quantum dynamics simulations to probe the behaviour of many-body systems. We first introduce a spectroscopy-based protocol to extract excitation spectra from the time evolution of observables and analyse its performance on near-term quantum devices. We show that the protocol requires only short evolution times and is resilient to errors, facilitating its implementation on real hardware. Building on this, we simulate fermionic excitation spectra using current quantum devices, introducing resource optimisations - including an approximate free-fermion initial state, a unitary quench without ancillae, and reduced Trotter steps - that significantly lower circuit depth. These improvements enable simulations at system sizes competitive with the state of the art, yielding results that match classical benchmarks. While most studies focus on unitary dynamics of isolated systems, open quantum dynamics is increasingly relevant for describing non-equilibrium processes across biology, materials science, and quantum control. We extend our scope to systems coupled to an environment and governed by Lindbladian evolution. By studying simulation complexity through physically meaningful quantities - such as correlation length and mixing time - we explore whether open-system dynamics can be as tractable as their unitary counterparts. These findings provide insights into the computational complexity landscape of open-system simulation and highlight regimes where quantum methods offer a clear advantage.
We draw a picture of physical systems that allows us to recognize what "time" is by requiring consistency with the way that time enters the fundamental laws of Physics. Elements of the picture are two non-interacting and yet entangled quantum systems, one of which acting as a clock. The setting is based on the Page and Wootters mechanism, with tools from large-N quantum approaches. Starting from an overall quantum description, we first take the classical limit of the clock only, and then of the clock and the evolving system altogether; we thus derive the Schrödinger equation in the first case, and the Hamilton equations of motion in the second. This work shows that there is not a "quantum time", possibly opposed to a "classical" one; there is only one time, and it is a manifestation of entanglement.
The last two decades feature an ever-increasing interest in quantum information processing - generalization of the classical computational models in the view of quantum mechanics (QM). In theory, quantum computational devices are capable of solving in polynomial time problems for which only algorithms with exponential time complexity are known. Such enormous advantage explains the global scale of scientific efforts for creating the quantum computer - physical realization of such computational device. While the results of those efforts are still constrained inside the large experimental laboratories, an adequate tool for studying quantum algorithms will be of great help for educating the next generation of computer scientists - the engineers that will be responsible for operating these devices. The paper describes a quantum computer simulation model that exhibits a certain attractive property - the simulation slowdown does not depend directly on the size of the input data but rather on the complexity of the quantum state, meaning that transformations upon less-entangled data are performed faster. Based on this property the simulation can be spread between the nodes of a grid cluster in a way as to keep the entanglement in each job minimal
We study how parallelism can speed up quantum simulation. A parallel quantum algorithm is proposed for simulating the dynamics of a large class of Hamiltonians with good sparse structures, called uniform-structured Hamiltonians, including various Hamiltonians of practical interest like local Hamiltonians and Pauli sums. Given the oracle access to the target sparse Hamiltonian, in both query and gate complexity, the running time of our parallel quantum simulation algorithm measured by the quantum circuit depth has a doubly (poly-)logarithmic dependence polyloglog(1/ϵ) on the simulation precision ϵ . This presents an exponential improvement over the dependence polylog(1/ϵ) of previous optimal sparse Hamiltonian simulation algorithm without parallelism. To obtain this result, we introduce a novel notion of parallel quantum walk, based on Childs' quantum walk. The target evolution unitary is approximated by a truncated Taylor series, which is obtained by combining these quantum walks in a parallel way. A lower bound Ω(loglog(1/ϵ)) is established, showing that the ϵ -dependence of the gate depth achieved in this work cannot be significantly improved.Our algorithm is applied to simulating three physical models: the Heisenberg model, the Sachdev-Ye-Kitaev model and a quantum chemistry model in second quantization. By explicitly calculating the gate complexity for implementing the oracles, we show that on all these models, the total gate depth of our algorithm has a polyloglog(1/ϵ) dependence in the parallel setting.
Quantum computers provide a fundamentally new computing paradigm that promises to revolutionize our ability to solve broad classes of problems. Surprisingly, the basic mathematical structures of gate-based quantum computing, such as unitary operations on a finite-dimensional Hilbert space, are not unique to quantum systems but may be found in certain classical systems as well. Previously, it has been shown that one can represent an arbitrary multi-qubit quantum state in terms of classical analog signals using nested quadrature amplitude modulated signals. Furthermore, using digitally controlled analog electronics one may manipulate these signals to perform quantum gate operations and thereby execute quantum algorithms. The computational capacity of a single signal is, however, limited by the required bandwidth, which scales exponentially with the number of qubits when represented using frequency-based encoding. To overcome this limitation, we introduce a method to extend this approach to multiple parallel signals. Doing so allows a larger quantum state to be emulated with the same gate time required for processing frequency-encoded signals. In the proposed representation, each doubling of the number of signals corresponds to an additional qubit in the spatial domain. Single quit gate operations are similarly extended so as to operate on qubits represented using either frequency-based or spatial encoding schemes. Furthermore, we describe a method to perform gate operations between pairs of qubits represented using frequency or spatial encoding or between frequency-based and spatially encoded qubits. Finally, we describe how this approach may be extended to represent qubits in the time domain as well.
The study of quantum many-body systems is crucial to our understanding of nature at the microscopic scale. As analytical solutions quickly become intractable, the development of efficient computational simulation techniques is indispensable to predict the behaviour of quantum systems. This task is particularly challenging as the dimension of the underlying Hilbert space grows exponentially with the system size. While classical computational methods have achieved remarkable success in specific regimes, their applicability is fundamentally limited by this exponential scaling. Quantum computers offer a natural platform for representing and simulating quantum systems, but current devices are constrained by noise and limited qubit counts, drastically restricting the algorithms that can be run. In this context, hybrid quantum-classical algorithms provide a promising approach to combine the strengths of both paradigms. This thesis develops hybrid quantum-classical algorithms for the simulation of quantum many-body systems. The key idea is to perform classically difficult tasks on near-term quantum devices, while leveraging classical computational power to control, correct, and extend the obtained results. Within this framework, we introduce two complementary approaches for the simulation of quantum dynamics. The first method employs circuit knitting techniques to distribute variational quantum simulations across multiple quantum devices. The entanglement between weakly coupled clusters of highly correlated spins is recovered in classical post-processing, where the overhead is controlled through a constrained optimisation problem. The second method combines a simplified, Trotterised quantum dynamics simulation with a parametrised classical ansatz trained to correct for the approximations made on the quantum device. We show that this method can be employed to correct Trotter errors, implement long-range gates with hardware-efficient circuits, or simulate larger systems by incorporating additional degrees of freedom purely classically. In addition, we present a hybrid approach to computing thermal properties of quantum many-body systems via the Quantum Finite Temperature Lanczos Method. A Krylov subspace is generated on a quantum computer by combining the efficient preparation of typical states with real-time evolution. A classical computer then diagonalises the system in the low-dimensional subspace and estimates the finite-temperature expectation values through trace estimation. The results presented in this thesis demonstrate that hybrid quantum-classical algorithms can expand the range of systems that can be studied on near-term quantum devices. Conversely, we show that incorporating quantum subroutines into classical simulation techniques can improve their efficiency. These results suggest that, rather than treating classical and quantum simulation as competing paradigms, their careful integration can provide a more effective route toward practical quantum many-body simulation.