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Register a new userA multiprecision matrix calculation library and its extension library for a matrix-product-state simulation of quantum computing
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Authors
Akira SaiToh
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A C++ library, named ZKCM, has been developed for the purpose of multiprecision matrix calculations, which is based on the GNU MP and MPFR libraries. It is especially convenient for writing programs involving tensor-product operations, tracing-out operations, and singular-value decompositions. Its extension library, ZKCM_QC, for simulating quantum computing has been developed using the time-dependent matrix-product-state simulation method. This report gives a brief introduction to the libraries with sample programs.
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The time-dependent matrix-product-state (TDMPS) simulation method has been used for numerically simulating quantum computing for a decade. We introduce our C++ library ZKCM_QC developed for multiprecision TDMPS simulations of quantum circuits. Besides its practical usability, the library is useful for evaluation of the method itself. With the library, we can capture two types of numerical errors in the TDMPS simulations: one due to rounding errors caused by the shortage in mantissa portions of floating-point numbers; the other due to truncations of nonnegligible Schmidt coefficients and their corresponding Schmidt vectors. We numerically analyze these errors in TDMPS simulations of quantum computing.
ZKCM is a C++ library developed for the purpose of multiprecision matrix computation, on the basis of the GNU MP and MPFR libraries. It provides an easy-to-use syntax and convenient functions for matrix manipulations including those often used in numerical simulations in quantum physics. Its extension library, ZKCM_QC, is developed for simulating quantum computing using the time-dependent matrix-product-state simulation method. This paper gives an introduction about the libraries with practical sample programs.
A quantum algorithm to simulate the real time dynamics of two-flavor massive Gross-Neveu model is presented in Schrodinger picture. We implement the simulation on a classic computer by applying the matrix product state representation. The real time evolutions of up to four particles on a site in initial state are figured out in space-time coordinate. The state evolutions are effectively affected by fermion mass and coupling constant of the model. Especially when the mass of fermion is small enough and the coupling is strong enough, the fundamental fermions evolve synchronistically in space from the two-fermion and four-fermion initial states. These are also the conditions on which the bound states made up of fundamental fermion pairs were found to arise automatically in the literatures.
We prove that ground states of gapped local Hamiltonians on an infinite spin chain can be efficiently approximated by matrix product states with a bond dimension which scales as D~(L-1)/epsilon, where any local quantity on L consecutive spins is approximated to accuracy epsilon.
Quantum trajectories and superoperator algorithms implemented within the matrix product state (MPS) framework are powerful tools to simulate the real-time dynamics of open dissipative quantum systems. As for the unitary case, the reachable time-scales as well as system sizes are limited by the (possible) build-up of entanglement entropy. The aforementioned methods constitute complementary approaches how Lindblad master equations can be integrated relying either on a quasi-exact representation of the full density matrix or a stochastic unraveling of the density matrix in terms of pure states. In this work, we systematically benchmark both methods by studying the dynamics of a Bose-Hubbard chain in the presence of local as well as global dephasing. The build-up as well as system-size scaling of entanglement entropy strongly depends on the method and the parameter regime and we discuss the applicability of the methods for these cases as well as study the distribution of observables and time discretization errors that can become a limiting factor for global dissipation.
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