[Paper Review] Hidden Quantum Markov Models and non-adaptive read-out of many-body states

Alex Monràs, Almut Beige+1 more|arXiv (Cornell University)|Feb 11, 2010

Quantum Computing Algorithms and Architecture23 references11 citations

TL;DR

This paper introduces Hidden Quantum Markov Models (HQMMs), a generalization of quantum finite-state generators that use stochastic quantum operations instead of von Neumann projections, enabling more efficient compression of stochastic processes. It demonstrates that HQMMs can generate process languages—such as alternating binary sequences—that classical Hidden Markov Models cannot, and shows a 1D cluster state's non-adaptive read-out is equivalent to a two-state HQMM, proving quantum models can achieve higher compression than classical ones.

ABSTRACT

Stochastic finite-state generators are compressed descriptions of infinite time series. Alternatively, compressed descriptions are given by quantum finite- state generators [K. Wiesner and J. P. Crutchfield, Physica D 237, 1173 (2008)]. These are based on repeated von Neumann measurements on a quantum dynamical system. Here we generalise the quantum finite-state generators by replacing the von Neumann pro jections by stochastic quantum operations. In this way we assure that any time series with a stochastic compressed description has a compressed quantum description. Moreover, we establish a link between our stochastic generators and the sequential readout of many-body states with translationally-invariant matrix product state representations. As an example, we consider the non-adaptive read-out of 1D cluster states. This is shown to be equivalent to a Hidden Quantum Model with two internal states, providing insight on the inherent complexity of the process. Finally, it is proven by example that the quantum description can have a higher degree of compression than the classical stochastic one.

Motivation & Objective

• To generalize quantum finite-state generators by replacing von Neumann projections with stochastic quantum operations, enabling broader class of process languages to be generated.
• To establish a formal link between HQMMs and the non-adaptive read-out of many-body quantum states with matrix product state (MPS) representations.
• To demonstrate that HQMMs can achieve higher compression efficiency than classical stochastic finite-state machines for certain process languages.
• To provide a quantum-theoretic framework for modeling sequential measurement processes with limited access, relevant to measurement-based quantum computation.
• To explore the role of quantum memory and non-projective measurements in reducing the number of internal states required for process generation.

Proposed method

• Replace projective (von Neumann) measurements in quantum finite-state generators with general quantum operations (completely positive maps), allowing non-deterministic and non-collapsing state evolution.
• Model the system as a dynamical open quantum system where each measurement outcome triggers a quantum operation, updating the system’s state based on the outcome.
• Use the Kraus operator formalism to represent the quantum operations, with each outcome associated with a Kraus operator $ K_y $, such that the post-measurement state is $ \rho' = \sum_y K_y \rho K_y^\dagger $.
• Define the process language as the sequence of classical measurement outcomes generated by repeated application of the quantum operations and measurements.
• Construct HQMMs for specific processes, such as alternating binary sequences, by designing appropriate Kraus operators that reproduce the desired outcome statistics.
• Establish equivalence between the non-adaptive read-out of a 1D cluster state and a two-level HQMM, showing that the measurement statistics match those of a repeatedly measured quantum system with specific quantum operations.

Experimental results

Research questions

• RQ1 Can stochastic quantum operations generalize quantum finite-state generators to model a broader class of stochastic processes than von Neumann measurements?
• RQ2 Is there a direct correspondence between the non-adaptive read-out of 1D cluster states and a quantum finite-state machine with generalized measurements?
• RQ3 Can HQMMs achieve a lower number of internal states than classical Hidden Markov Models for the same process language?
• RQ4 What is the role of quantum memory and non-projective measurements in enabling more efficient compression of stochastic processes?
• RQ5 How does the dimensionality of the underlying quantum system constrain the class of process languages that can be generated?

Key findings

• HQMMs generalize quantum finite-state generators by allowing non-projective, stochastic quantum operations, enabling the generation of process languages that are inaccessible to von Neumann-based models.
• The non-adaptive read-out of a 1D cluster state is equivalent to a two-state Hidden Quantum Markov Model, demonstrating a direct physical realization of an HQMM in a many-body system.
• A specific alternating binary sequence with unequal probabilities cannot be generated by a two-state Hidden Markov Model but is successfully generated by a two-state HQMM using non-projective measurements.
• The HQMM representation for the alternating sequence uses Kraus operators $ K_0 = \frac{1}{\sqrt{2}}|\uparrow\rangle\langle\uparrow| $, $ K_1 = \frac{1}{\sqrt{2}}|\downarrow\rangle\langle\downarrow| $, $ K_2 = \frac{1}{\sqrt{2}}|+\rangle\langle+| $, $ K_3 = \frac{1}{\sqrt{2}}|-\rangle\langle-| $, which reproduce the correct outcome statistics.
• The model described by the transition matrix in Eq. (75) cannot be implemented by a two-state HMM, but is realizable via the HQMM in Eq. (88), proving a quantum advantage in compression.
• The paper proves by example that quantum descriptions can achieve higher compression than classical stochastic models, as the HQMM uses fewer internal states than the minimal classical HMM for the same process.