Lower bounds for ground states of condensed matter systems

Baumgratz, Tillmann; Plenio, Martin B.

doi:10.1088/1367-2630/14/2/023027

Cited by 34 publications

(39 citation statements)

References 29 publications

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“…Unique to this program is the sparsity of each constraint relative to the total number of variables in the program. A class of SDP solvers using the augmented Lagrangian technique have been shown to efficiently solve SDPs of this form in quantum chemistry and condensed matter [82,85,87,[114][115][116][117]. The primal SDP is mathematically stated as á ñ ( ) C X min , D1…”

Section: Appendix D Computational Implementation Of the Reconstructimentioning

confidence: 99%

Application of fermionic marginal constraints to hybrid quantum algorithms

2018

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Many quantum algorithms, including recently proposed hybrid classical/quantum algorithms, make use of restricted tomography of the quantum state that measures the reduced density matrices, or marginals, of the full state. The most straightforward approach to this algorithmic step estimates each component of the marginal independently without making use of the algebraic and geometric structure of the marginals. Within the field of quantum chemistry, this structure is termed the fermionic n-representability conditions, and is supported by a vast amount of literature on both theoretical and practical results related to their approximations. In this work, we introduce these conditions in the language of quantum computation, and utilize them to develop several techniques to accelerate and improve practical applications for quantum chemistry on quantum computers. As a general result, we demonstrate how these marginals concentrate to diagonal quantities when measured on random quantum states. We also show that one can use fermionic n-representability conditions to reduce the total number of measurements required by more than an order of magnitude for medium sized systems in chemistry. As a practical demonstration, we simulate an efficient restoration of the physicality of energy curves for the dilation of a four qubit diatomic hydrogen system in the presence of three distinct one qubit error channels, providing evidence these techniques are useful for pre-fault tolerant quantum chemistry experiments. semidefinite-ness, constrained spin-, and particle-numbers are expected to increase the observed energy of the 2-RDM with respect to the uncorrected noisy 2-RDM.

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Section: Appendix D Computational Implementation Of the Reconstructimentioning

confidence: 99%

Application of fermionic marginal constraints to hybrid quantum algorithms

2018

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“…Direct calculation of the reduced variables, however, requires that they and their functionals be consistent with a realistic N -electron quantum system; in other words, the reduced variables and functionals must be representable by the integration of an N -electron density matrix. Such consistency relations are known as the N -representability conditions [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][20][21][22][23][24]. These conditions are particularly important to 2-RDM methods where they enable the direct calculation of the 2-RDM without the wavefunction, but they are also implicit in the design of realistic approximations to the density functional in density functional theory [31,32].…”

Section: Introductionmentioning

confidence: 99%

Significant conditions for the two-electron reduced density matrix from the constructive solution ofNrepresentability

Mazziotti

2012

Phys. Rev. A

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We recently presented a constructive solution to the N -representability problem of the twoelectron reduced density matrix (2-RDM)-a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic N -electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. 108, 263002 (2012)]. In this paper we provide additional details and derive further N -representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2, q)-positivity conditions where the q indicates their derivation from the nonnegativity of q-body operators. In addition to the known T1 and T2 conditions, we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is conditions which arise from lifting lower (2, q)-positivity conditions to higher (2, q + 1)-positivity conditions and (ii) pure conditions, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2, q)-positivity conditions for q > 3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the new N -representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation.

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“…(1) was unknown until recently, 8 an approximate class of Nrepresentability constraints has been demonstrated to be sufficient for calculating ground-state properties of the metal-to-insulator transition of hydrogen chains, 9 ground states and charge distributions of quantum dots, 10 quantum phase transitions, 11,12 dissociation channels, 13 and quantum lattice systems. [14][15][16][17][18][19][20] Variational minimization of the energy as a functional of the 2-RDM is expressible as a special convex optimization problem known as a semidefinite program.…”

Section: Introductionmentioning

confidence: 99%

Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduceddensity- matrix method

Rubin¹,

Mazziotti²

2014

Highlights in Theoretical Chemistry

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Minimizing the energy of an N -electron system as a functional of a two-electron reduced density matrix (2-RDM), constrained by necessary N -representability conditions (conditions for the 2-RDM to represent an ensemble N -electron quantum system), yields a rigorous lower bound to the groundstate energy in contrast to variational wavefunction methods. We characterize the performance of two sets of approximate constraints, (2,2)-positivity (DQG) and approximate (2,3)-positivity (DQGT) conditions, at capturing correlation in one-dimensional and quasi-one-dimensional (ladder) Hubbard models. We find that, while both the DQG and DQGT conditions capture both the weak and strong correlation limits, the more stringent DQGT conditions improve the ground-state energies, the natural occupation numbers, the pair correlation function, the effective hopping, and the connected (cumulant) part of the 2-RDM. We observe that the DQGT conditions are effective at capturing strong electron correlation effects in both one-and quasi-one-dimensional lattices for both half filling and less-than-half filling.

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Lower bounds for ground states of condensed matter systems

Cited by 34 publications

References 29 publications

Application of fermionic marginal constraints to hybrid quantum algorithms

Application of fermionic marginal constraints to hybrid quantum algorithms

Significant conditions for the two-electron reduced density matrix from the constructive solution ofNrepresentability

Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduceddensity- matrix method

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