A Semidefinite Programming algorithm for the Quantum Mechanical Bootstrap

• URL: http://arxiv.org/abs/2209.14332v1
• Date: Wed, 28 Sep 2022 18:02:58 GMT
• Title: A Semidefinite Programming algorithm for the Quantum Mechanical Bootstrap
• Authors: David Berenstein, George Hulsey
• Abstract summary: We present a semidefinite program (SDP) algorithm to find eigenvalues of Schr"odinger operators within the bootstrap approach to quantum mechanics.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a semidefinite program (SDP) algorithm to find eigenvalues of Schr\"{o}dinger operators within the bootstrap approach to quantum mechanics. The bootstrap approach involves two ingredients: a nonlinear set of constraints on the variables (expectation values of operators in an energy eigenstate), plus positivity constraints (unitarity) that need to be satisfied. By fixing the energy we linearize all the constraints and show that the feasability problem can be presented as an optimization problem for the variables that are not fixed by the constraints and one additional slack variable that measures the failure of positivity. To illustrate the method we are able to obtain high-precision, sharp bounds on eigenenergies for arbitrary confining polynomial potentials in 1-D.

Related papers

• Certifying steady-state properties of open quantum systems [0.0]
  Estimating steady state properties of open quantum systems is a crucial problem in quantum technology. We show how to derive in a scalable way using semi-definite programming certified bounds on the expectation value of an arbitrary observable.
  arXiv  Detail & Related papers  (2024-10-17T15:13:12Z)
• Inequality constraints in variational quantum circuits with qudits [1.0923877073891446]
  Quantum optimization is emerging as a prominent candidate for exploiting the capabilities of near-term quantum devices. Here, we study an alternative direct implementation of inequality constraints within the QAOA algorithm, achieved using qudit-SUM gates. We find that the direct implementation of the inequality penalties vastly outperforms the slack variables method, especially when studying real-world inspired problems with many constraints.
  arXiv  Detail & Related papers  (2024-10-10T07:32:38Z)
• Uncertainty relation and the constrained quadratic programming [3.397254718930225]
  We find that the tight state-independent lower bound of the variance sum can be characterized as a quadratic programming problem with nonlinear constraints in optimization theory. We introduce a numerical algorithm tailored for solving these quadratic programming instances, highlighting its efficiency and accuracy.
  arXiv  Detail & Related papers  (2024-04-29T13:11:20Z)
• Variational-quantum-eigensolver-inspired optimization for spin-chain work extraction [39.58317527488534]
  Energy extraction from quantum sources is a key task to develop new quantum devices such as quantum batteries. One of the main issues to fully extract energy from the quantum source is the assumption that any unitary operation can be done on the system. We propose an approach to optimize the extractable energy inspired by the variational quantum eigensolver (VQE) algorithm.
  arXiv  Detail & Related papers  (2023-10-11T15:59:54Z)
• Scalable Bayesian Meta-Learning through Generalized Implicit Gradients [64.21628447579772]
  Implicit Bayesian meta-learning (iBaML) method broadens the scope of learnable priors, but also quantifies the associated uncertainty. Analytical error bounds are established to demonstrate the precision and efficiency of the generalized implicit gradient over the explicit one.
  arXiv  Detail & Related papers  (2023-03-31T02:10:30Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• Unbalanced penalization: A new approach to encode inequality constraints of combinatorial problems for quantum optimization algorithms [42.29248343585333]
  We present an alternative method that does not require extra slack variables. We evaluate our approach on the traveling salesman problem, the bin packing problem, and the knapsack problem. This new approach can be used to solve problems with inequality constraints with a reduced number of resources.
  arXiv  Detail & Related papers  (2022-11-25T06:05:18Z)
• Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints [62.72309460291971]
  We introduce a variational quantum algorithm for Goemans-Williamson algorithm that uses only $n+1$ qubits. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems.
  arXiv  Detail & Related papers  (2022-06-30T03:15:23Z)
• Q-FW: A Hybrid Classical-Quantum Frank-Wolfe for Quadratic Binary Optimization [44.96576908957141]
  We present a hybrid classical-quantum framework based on the Frank-Wolfe algorithm, Q-FW, for solving quadratic, linear iterations problems on quantum computers.
  arXiv  Detail & Related papers  (2022-03-23T18:00:03Z)
• A Stochastic Composite Augmented Lagrangian Method For Reinforcement Learning [9.204659134755795]
  We consider the linear programming (LP) formulation for deep reinforcement learning. The augmented Lagrangian method suffers the double-sampling obstacle in solving the LP. A deep parameterized augment Lagrangian method is proposed.
  arXiv  Detail & Related papers  (2021-05-20T13:08:06Z)
• Preparing Bethe Ansatz Eigenstates on a Quantum Computer [0.0]
  We present a quantum algorithm for preparing Bethe ansatz eigenstates of the spin-1/2 XXZZ spin chain that correspond to real-valued solutions of the Bethe equations. Although the algorithm is probabilistic, with a success rate that decreases with increasing eigenstate energy, we employ amplification to boost the success probability.
  arXiv  Detail & Related papers  (2021-03-24T17:58:21Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.