NLTS Hamiltonians from good quantum codes

• URL: http://arxiv.org/abs/2206.13228v3
• Date: Sun, 28 Jul 2024 22:16:51 GMT
• Title: NLTS Hamiltonians from good quantum codes
• Authors: Anurag Anshu, Nikolas P. Breuckmann, Chinmay Nirkhe,
• Abstract summary: The NLTS (No Low-Energy Trivial State) conjecture posits that there exist families of Hamiltonians with all low energy states of non-trivial complexity. We prove this conjecture by showing that the recently discovered families of constant-rate and linear-distance QLDPC codes correspond to NLTS local Hamiltonians.
• Score: 14.00987234726578
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings [2014] posits that there exist families of Hamiltonians with all low energy states of non-trivial complexity (with complexity measured by the quantum circuit depth preparing the state). We prove this conjecture by showing that the recently discovered families of constant-rate and linear-distance QLDPC codes correspond to NLTS local Hamiltonians.

Related papers

• Complexity of geometrically local stoquastic Hamiltonians [1.474723404975345]
  The QMA-completeness of the local Hamiltonian problem is a landmark result of the field of Hamiltonian complexity. We show that both the two- and one-dimensional geometrically local analogues remain MA-hard with high enough qudit dimension.
  arXiv  Detail & Related papers  (2024-07-22T09:27:25Z)
• Coherence generation with Hamiltonians [44.99833362998488]
  We explore methods to generate quantum coherence through unitary evolutions. This quantity is defined as the maximum derivative of coherence that can be achieved by a Hamiltonian. We identify the quantum states that lead to the largest coherence derivative induced by the Hamiltonian.
  arXiv  Detail & Related papers  (2024-02-27T15:06:40Z)
• Local Hamiltonians with no low-energy stabilizer states [0.4588028371034407]
  A family of local Hamiltonians with low-enough constant energy do not have succinct representations allowing perfect sampling access. We describe families that exhibit this requisite property via a simple alteration to local Hamiltonians corresponding to CSS codes. Our method can also be applied to the recent NLTS Hamiltonians of Anshu, Breuckmann, and Nirkhe [ABN22], resulting in a family of local Hamiltonians whose low-energy space contains neither stabilizer states nor trivial states.
  arXiv  Detail & Related papers  (2023-02-28T16:55:05Z)
• Sparse random Hamiltonians are quantumly easy [105.6788971265845]
  A candidate application for quantum computers is to simulate the low-temperature properties of quantum systems. This paper shows that, for most random Hamiltonians, the maximally mixed state is a sufficiently good trial state. Phase estimation efficiently prepares states with energy arbitrarily close to the ground energy.
  arXiv  Detail & Related papers  (2023-02-07T10:57:36Z)
• Symmetric Pruning in Quantum Neural Networks [111.438286016951]
  Quantum neural networks (QNNs) exert the power of modern quantum machines. QNNs with handcraft symmetric ansatzes generally experience better trainability than those with asymmetric ansatzes. We propose the effective quantum neural tangent kernel (EQNTK) to quantify the convergence of QNNs towards the global optima.
  arXiv  Detail & Related papers  (2022-08-30T08:17:55Z)
• Improved Hardness Results for the Guided Local Hamiltonian Problem [1.53934570513443]
  Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. We show that the BQP-completeness persists even with 2-local Hamiltonians. We also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians.
  arXiv  Detail & Related papers  (2022-07-21T01:13:32Z)
• A construction of Combinatorial NLTS [22.539300644593936]
  NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings [2014] posits that there exist families of Hamiltonians with all low energy states of high complexity. Here, we prove a weaker version called the NLTS, where a quantum circuit lower bound is shown against states that violate a (small) constant fraction of local terms.
  arXiv  Detail & Related papers  (2022-06-06T16:55:34Z)
• Theory of Quantum Generative Learning Models with Maximum Mean Discrepancy [67.02951777522547]
  We study learnability of quantum circuit Born machines (QCBMs) and quantum generative adversarial networks (QGANs) We first analyze the generalization ability of QCBMs and identify their superiorities when the quantum devices can directly access the target distribution. Next, we prove how the generalization error bound of QGANs depends on the employed Ansatz, the number of qudits, and input states.
  arXiv  Detail & Related papers  (2022-05-10T08:05:59Z)
• Algebraic Compression of Quantum Circuits for Hamiltonian Evolution [52.77024349608834]
  Unitary evolution under a time dependent Hamiltonian is a key component of simulation on quantum hardware. We present an algorithm that compresses the Trotter steps into a single block of quantum gates. This results in a fixed depth time evolution for certain classes of Hamiltonians.
  arXiv  Detail & Related papers  (2021-08-06T19:38:01Z)
• Circuit lower bounds for low-energy states of quantum code Hamiltonians [17.209060627291315]
  We prove circuit lower bounds for all low-energy states of local Hamiltonians arising from quantum error-correcting codes. We show that low-depth states cannot accurately approximate the ground-energy even in physically relevant systems.
  arXiv  Detail & Related papers  (2020-11-03T22:36:22Z)
• On the Hamiltonian for three bosons with point interactions [77.34726150561087]
  We briefly summarize the most relevant steps in the search of rigorous results about the properties of quantum systems made of three bosons interacting with zero-range forces. We also describe recent attempts to solve the unboundedness problem of point-interaction Hamiltonians for a three-boson system, keeping the spectrum structure at low energies.
  arXiv  Detail & Related papers  (2020-01-28T16:52:04Z)

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.