New random compiler for Hamiltonians via Markov Chains

• URL: http://arxiv.org/abs/2411.06485v1
• Date: Sun, 10 Nov 2024 14:57:25 GMT
• Title: New random compiler for Hamiltonians via Markov Chains
• Authors: Benoît Dubus, Jérémie Roland,
• Abstract summary: Many quantum algorithms, such as adiabatic algorithms, require simulating Hamiltonian evolution. We develop a new compiler, similar to the first order randomized Trotter, but with an arguably simpler framework. It is more versatile as it supports a large class of randomisation schemes and as well as time-dependent weights.
• Score: 0.08192907805418585
• License:
• Abstract: Many quantum algorithms, such as adiabatic algorithms (\textit{e.g.} AQC) and phase randomisation, require simulating Hamiltonian evolution. In addition, the simulation of physical systems is an important objective in its own right. In many cases, the Hamiltonian is complex at first sight, but can be decomposed as a linear combination of simple ones; for instance, a sum of local Hamiltonians for Ising models or a sum of time-independent Hamiltonians with time-dependent coefficients (which is typically the case for adiabatic algorithms). In this paper we develop a new compiler, similar to the first order randomized Trotter, or qDRIFT~\cite{campbellRandomCompilerFast2019}, but with an arguably simpler framework. It is more versatile as it supports a large class of randomisation schemes and as well as time-dependent weights. We first present the model and derive its governing equations. We then define and analyze the simulation error for a sum of two Hamiltonians, and generalize it to a sum of $Q$ Hamiltonians. We prove that the number of gates necessary to simulate the weighted sum of $Q$ Hamiltonians of magnitude $C$ during a time $T$ with an error less than $\epsilon_0$ grows as $\tilde{\mathcal{O}}\left(C^2T^2\epsilon_0^{-1}\right)$.

Related papers

• Slow Mixing of Quantum Gibbs Samplers [47.373245682678515]
  We present a quantum generalization of these tools through a generic bottleneck lemma. This lemma focuses on quantum measures of distance, analogous to the classical Hamming distance but rooted in uniquely quantum principles. Even with sublinear barriers, we use Feynman-Kac techniques to lift classical to quantum ones establishing tight lower bound $T_mathrmmix = 2Omega(nalpha)$.
  arXiv  Detail & Related papers  (2024-11-06T22:51:27Z)
• Predicting Ground State Properties: Constant Sample Complexity and Deep Learning Algorithms [48.869199703062606]
  A fundamental problem in quantum many-body physics is that of finding ground states of local Hamiltonians. We introduce two approaches that achieve a constant sample complexity, independent of system size $n$, for learning ground state properties.
  arXiv  Detail & Related papers  (2024-05-28T18:00:32Z)
• Simplifying the simulation of local Hamiltonian dynamics [0.0]
  Local Hamiltonians, $H_k$, describe non-trivial $k$-body interactions in quantum many-body systems. We build upon known methods to derive examples of $H_k$ and $H_k'$ that simulate the same physics. We propose a method to search for the $k'$-local Hamiltonian that simulates, with the highest possible precision, the short time dynamics of a given $H_k$ Hamiltonian.
  arXiv  Detail & Related papers  (2023-10-10T22:31:45Z)
• Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians [65.268245109828]
  We develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations. We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix. A quantum algorithm to simulate the dynamics of the density matrix is proposed.
  arXiv  Detail & Related papers  (2023-06-14T23:08:51Z)
• On the Impossibility of General Parallel Fast-forwarding of Hamiltonian Simulation [4.925967492198012]
  Hamiltonian simulation is one of the most important problems in the field of quantum computing. Existing simulation algorithms require running time at least linear in the evolution time $T$. It is intriguing whether we can achieve fast Hamiltonian simulation with the power of parallelism.
  arXiv  Detail & Related papers  (2023-05-21T12:30:00Z)
• Unbiased random circuit compiler for time-dependent Hamiltonian simulation [8.694056486825318]
  Time-dependent Hamiltonian simulation is a critical task in quantum computing. We develop an unbiased random compiler for TDHS. We perform numerical simulations for a spin model under the interaction picture and the adiabatic ground state preparation for molecular systems.
  arXiv  Detail & Related papers  (2022-12-19T13:40:05Z)
• Average-case Speedup for Product Formulas [69.68937033275746]
  Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states. Our results open doors to the study of quantum algorithms in the average case.
  arXiv  Detail & Related papers  (2021-11-09T18:49:48Z)
• Hamiltonian simulation with random inputs [74.82351543483588]
  Theory of average-case performance of Hamiltonian simulation with random initial states. Numerical evidence suggests that this theory accurately characterizes the average error for concrete models.
  arXiv  Detail & Related papers  (2021-11-08T19:08:42Z)
• 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)
• Quantum algorithm for time-dependent Hamiltonian simulation by permutation expansion [6.338178373376447]
  We present a quantum algorithm for the dynamical simulation of time-dependent Hamiltonians. We demonstrate that the cost of the algorithm is independent of the Hamiltonian's frequencies.
  arXiv  Detail & Related papers  (2021-03-29T05:02:02Z)
• Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation [0.8057006406834467]
  We present a quantum algorithm to solve systems of linear equations of the form $Amathbfx=mathbfb$. The algorithm achieves an exponential improvement with respect to $N$ over classical methods.
  arXiv  Detail & Related papers  (2020-09-09T18:00:09Z)

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.