Note on the bounds and approximations to the ground state energy of N-fermion systems derived from density matrix theory

1970 ◽  
Vol 48 (2) ◽  
pp. 147-149 ◽  
Author(s):  
F. David Peat
Keyword(s):  
Lower Bound ◽  
Density Matrix ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Matrix Theory ◽  
Fermion Systems ◽  
Reduced Density ◽  
Density Matrix Theory ◽  
Exact Energy

Certain lower bounds to the ground state energy of N-fermion systems have been derived in the literature using the properties of reduced density matrices.It is indicated that while some good approximations to the energy may be obtained by similar consideration the rigorous lower bound expressions lie too far below the exact energy to prove generally useful.

scholarly journals Quantum simulation of molecules without fermionic encoding of the wave function

2021 ◽  
Vol 23 (11) ◽  
pp. 113037
Author(s):  
David A Mazziotti ◽  
Scott E Smart ◽  
Alexander R Mazziotti
Keyword(s):  
Wave Function ◽  
Density Matrix ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Molecular Simulations ◽  
Wave Functions ◽  
Reduced Density Matrix ◽  
Particle Wave Function ◽  
Reduced Density

Abstract Molecular simulations generally require fermionic encoding in which fermion statistics are encoded into the qubit representation of the wave function. Recent calculations suggest that fermionic encoding of the wave function can be bypassed, leading to more efficient quantum computations. Here we show that the two-electron reduced density matrix (2-RDM) can be expressed as a unique functional of the unencoded N-qubit-particle wave function without approximation, and hence, the energy can be expressed as a functional of the 2-RDM without fermionic encoding of the wave function. In contrast to current hardware-efficient methods, the derived functional has a unique, one-to-one (and onto) mapping between the qubit-particle wave functions and 2-RDMs, which avoids the over-parametrization that can lead to optimization difficulties such as barren plateaus. An application to computing the ground-state energy and 2-RDM of H4 is presented.

The Schrödinger–Riccati equation. The ground-state energy of Be I

2002 ◽  
Vol 80 (9) ◽  
pp. 1053-1057 ◽  
Author(s):  
S Fraga ◽  
JM García de la Vega ◽  
E S Fraga
Keyword(s):  
Riccati Equation ◽  
Relative Error ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Maximum Relative Error ◽  
Statistical Calculation ◽  
Absolute Value ◽  
A Value ◽  
Exact Energy

The Schrödinger–Riccati equation has been used for the prediction of the ground-state energy of Be I. A statistical calculation yields a value of –14.670 hartree, with a maximum relative error of 0.02% (in absolute value) with respect to the exact energy of –14.667 36 hartree. PACS Nos.: 31.25Eb, 31.10+z, 02.70-c, 31.15Bs

Energy bounds for the spiked harmonic oscillator

1995 ◽  
Vol 73 (7-8) ◽  
pp. 493-496 ◽  
Author(s):  
Richard L. Hall ◽  
Nasser Saad
Keyword(s):  
Lower Bound ◽  
Harmonic Oscillator ◽  
Ground State Energy ◽  
Upper Bound ◽  
Ground State ◽  
Trial Function ◽  
State Energy ◽  
Parameter Range ◽  
Complementary Energy ◽  
Energy Bounds

A three-parameter variational trial function is used to determine an upper bound to the ground-state energy of the spiked harmonic-oscillator Hamiltonian [Formula: see text]. The entire parameter range λ > 0 and α ≥ 1 is treated in a single elementary formulation. The method of potential envelopes is also employed to derive a complementary energy lower bound formula valid for all the discrete eigenvalues.

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