The Thermodynamic Cost of Ignorance: Thermal State Preparation with One Ancilla Qubit

• URL: http://arxiv.org/abs/2502.03410v1
• Date: Wed, 05 Feb 2025 17:50:37 GMT
• Title: The Thermodynamic Cost of Ignorance: Thermal State Preparation with One Ancilla Qubit
• Authors: Matthew Hagan, Nathan Wiebe,
• Abstract summary: We investigate a model of thermalization wherein a single ancillary qubit randomly interacts with the system to be thermalized. This not only sheds light on the emergence of Gibbs states in nature, but also provides a routine for preparing arbitrary thermal states on a digital quantum computer.
• Score: 0.5729426778193399
• License:
• Abstract: In this work we investigate a model of thermalization wherein a single ancillary qubit randomly interacts with the system to be thermalized. This not only sheds light on the emergence of Gibbs states in nature, but also provides a routine for preparing arbitrary thermal states on a digital quantum computer. For desired $\beta$ and random interaction $G$ the routine boils down to time independent Hamiltonian simulation and is represented by the channel $\Phi : \rho \mapsto \mathbb{E}_G {\rm Tr}_{\rm Env} \left[ e^{-i(H + \alpha G)t} \left(\rho \otimes \frac{e^{-\beta H_E}}{\mathcal{Z}}\right) e^{i (H + \alpha G)t} \right]$. We rigorously prove that these dynamics reduce to a Markov chain process in the weak-coupling regime with the thermal state as the approximate fixed point. We upper bound the total simulation time required in terms of the Markov chain spectral gap $\lambda_\star$, which we compute exactly in the ground state limit. These results are independent of any eigenvalue knowledge of the system, but we are further able to show that with knowledge of eigenvalue differences $\lambda_S(i) - \lambda_S(j)$, then the total simulation time is dramatically reduced. The ratio of the complete ignorance simulation cost to the perfect knowledge simulation cost scales as $\widetilde{O} \left({\frac{\|{H_S}\|^7}{\delta_{\rm min}^7 \epsilon^{3.5} \lambda_\star(\beta)^{3.5}}}\right)$, where $\delta_{\min}$ is related to the eigenvalue differences of the system. Additionally, we provide more specific results for single qubit and harmonic oscillator systems as well as numeric experiments with hydrogen chains. In addition to the algorithmic merits, these results can be viewed as broad extensions of the Repeated Interactions model to generic Hamiltonians with unknown interactions, giving a complete picture of the thermalization process for quantum systems.

Related papers

• Exact Solvability Of Entanglement For Arbitrary Initial State in an Infinite-Range Floquet System [0.5371337604556311]
  We introduce an $N$-spin Floquet model with infinite-range Ising interactions. We numerically show that the values $langle Srangle/S_Max rightarrow 1$ for Ising strength deviates from $1$ for arbitrary initial states even though the thermodynamic limit does not exist in our model.
  arXiv  Detail & Related papers  (2024-11-25T18:55:05Z)
• 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. We show how to lift classical slow mixing results in the presence of a transverse field using Poisson Feynman-Kac techniques.
  arXiv  Detail & Related papers  (2024-11-06T22:51:27Z)
• Calculating response functions of coupled oscillators using quantum phase estimation [40.31060267062305]
  We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer. Our proposed quantum algorithm operates in the standard $s-sparse, oracle-based query access model. We show that a simple adaptation of our algorithm solves the random glued-trees problem in time.
  arXiv  Detail & Related papers  (2024-05-14T15:28:37Z)
• Hamiltonian simulation for low-energy states with optimal time dependence [45.02537589779136]
  We consider the task of simulating time evolution under a Hamiltonian $H$ within its low-energy subspace. We present a quantum algorithm that uses $O(tsqrtlambdaGamma + sqrtlambda/Gammalog (1/epsilon))$ queries to the block-encoding for any $Gamma$.
  arXiv  Detail & Related papers  (2024-04-04T17:58:01Z)
• Thermalization of closed chaotic many-body quantum systems [0.0]
  We investigate thermalization of a chaotic many-body quantum system by combining the Hartree-Fock approach and the Bohigas-Giannoni-Schmit conjecture. We show that in the semiclassical regime, $rm Tr (A rho(t))$ decays with time scale $hbar / Delta$ towards an equilibrium value.
  arXiv  Detail & Related papers  (2023-10-04T11:16:35Z)
• Weak universality, quantum many-body scars and anomalous infinite-temperature autocorrelations in a one-dimensional spin model with duality [0.0]
  We study a one-dimensional spin-$1/2$ model with three-spin interactions and a transverse magnetic field $h$. We compute the critical exponents $z$, $beta$, $gamma$ and $nu$, and the central charge $c$. For a system with periodic boundary conditions, there are an exponentially large number of exact mid-spectrum zero-energy eigenstates.
  arXiv  Detail & Related papers  (2023-07-20T18:00:05Z)
• Random-Matrix Model for Thermalization [0.0]
  Isolated quantum system said to thermalize if $rm Tr (A rho(t)) to rm Tr (A rho_rm eq)$ for time. $rho_rm eq(infty)$ is the time-independent density matrix.
  arXiv  Detail & Related papers  (2022-11-22T10:43:29Z)
• On parametric resonance in the laser action [91.3755431537592]
  We consider the selfconsistent semiclassical Maxwell--Schr"odinger system for the solid state laser. We introduce the corresponding Poincar'e map $P$ and consider the differential $DP(Y0)$ at suitable stationary state $Y0$.
  arXiv  Detail & Related papers  (2022-08-22T09:43:57Z)
• Quantum algorithms from fluctuation theorems: Thermal-state preparation [0.09786690381850353]
  We present a quantum algorithm to prepare a purification of the thermal state of $H_$ at inverse temperature. The dependence of the complexity in $epsilon$ varies according to the structure of the quantum systems. We analyze the complexity for preparing the thermal state of the transverse field Ising model using different non-equilibrium unitary processes.
  arXiv  Detail & Related papers  (2022-03-16T18:55:12Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)
• Quantum Algorithms for Simulating the Lattice Schwinger Model [63.18141027763459]
  We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In lattice units, we find a Schwinger model on $N/2$ physical sites with coupling constant $x-1/2$ and electric field cutoff $x-1/2Lambda$. We estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable---the mean pair density.
  arXiv  Detail & Related papers  (2020-02-25T19:18:36Z)

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.