Quantum Information Seminar | September 23, 16:00
Approximate Strong Simulation of 1D Quantum Dynamics with Reduced Transition Matrix Truncation
Tensor networks make the classical simulation of quantum many-body systems possible by restricting the description to the weakly entangled region of the Hilbert space. For the real-time dynamics of one-dimensional chains, the standard algorithm is the Time-Evolving Block Decimation (TEBD), which compresses the evolved state as a matrix product state. Its cost is set by the spatial entanglement, which after a global quench grows linearly in time: the required bond dimension grows exponentially, and the simulation is confined to short times. This is the entanglement barrier. This work addresses the problem of the approximate strong simulation of the dynamics: the computation of a single amplitude ⟨x|U(T)|y⟩ between two product states, to a relative precision fixed in advance. Instead of contracting the spatio-temporal network along the time direction, as in TEBD, we contract it along the spatial direction, a strategy known as transverse contraction. The truncation is based on the reduced transition matrix (RTM), built from the two halves of the network to optimise their overlap, and the cost of the contraction is measured by the entropy of the RTM singular-value spectrum. The algorithm was implemented from scratch in Julia and tested on kicked Ising and random brickwork circuits, and on the Trotterised mixed-field Ising chain in the gapped, critical and chaotic regimes. For circuits, we find numerical evidence that the RTM contraction requires subexponentially growing resources over the accessible time window, at finite relative precision. We relate this behaviour to the exponential decay of the retained RTM singular values, and to the finite precision, which restricts the information that must be retained. For Hamiltonian evolution, we examine the cost in the three regimes, for chains of 20 to 80 sites. It is largest at the critical point and smallest in the gapped chain. The accessible window is much shorter here and does not allow the growth law to be determined. This approach may offer a route to the probability statistics of chaotic quantum states, and to the benchmarking of quantum circuits, without constructing the full wave function.
University of Bologna
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Contact: Xhek Turkeshi