Operator dynamics in k-Markov random circuits

The gist: k-Markov sequences of unitary gates provide low-cost handles to manipulate the rate and structure of information spreading compared to traditional random, 0-Markov, circuits.

Introduction to k-Markov Sequences

k-Markov sequences of unitary gates allow for the manipulation of information dynamics by making the probability of the next gate depend on a fixed number of previous letters in a sequence, where k > 1 1, the sequences are k-Markov, often also called non-Markovian with context k [9]". In contrast to most studies of random circuits to date which assume k = 0 or introduce temporal structure through measurement and feedback <ref:2603.1821758, In contrast to most studies of random circuits to date assume k = 0, or introduce temporal structure through measurement and feedback [10–12]. Here the paper shows that k-Markov sequences provide operationally inexpensive dials to (statistically) control information dynamics without requiring mid-circuit disturbance of the state or real-time adjustment of the algorithm <ref:2603.1821759, Here we show that k-Markov sequences provide operationally inexpensive dials to (statistically) control information dynamics without requiring mid-circuit disturbance of the state or real-time adjustment of the algorithm.

Setting Up Circuits and Characterizing Dynamics

The paper sets up circuits using an alphabet of two letters (A, B), each defined by a particular choice of N-qubit unitary dynamics built from 2-local unitaries applied to nearest neighbor qubits in a 1-dimensional, closed chain <ref:2603.182171, We consider an alphabet of two letters (A, B), each defined by a particular choice of N-qubit unitary dynamics built from 2-local unitaries applied to nearest neighbor qubits in a 1-dimensional, closed chain. For example, 1-Markov sequences are defined by two transition probabilities <ref:2603.182172, For example, 1-Markov sequences are defined by two transition probabilities. A 2-Markov rule is specified by four probabilities <ref:2603.182174, a 2-Markov rule is specified by four probabilities. To characterize the dynamics produced by the k-Markov gate sequences, researchers examine properties of O(l) = U†l O(0)Ul where Ul is the cumulative unitary evolution of the circuit through layer l and O(0) to be an operator with a non-identity component on only a single site <ref:2603.182175, "To characterize the dynamics produced by the k-Markov gate sequences, we examine properties of O(l) = U†l O(0)Ul, where Ul is the cumulative unitary evolution of the circuit through layer l and O(0) to be an operator with a non-identity component on only a single site". The Pauli words, Pα, are used as the basis for operators and written as O(l) = Pα cα(l)Pα <ref:2603.182175, We will use the Pauli words, Pα, as the basis for operators and write O(l) = U†l O(0)Ul = P α c α (l)P α.

Operator Spreading and Cover Time

The relationship between k-Markov sequences and operator spreading is demonstrated in a simple case of only SWAP gates <ref:2603.182176, We first demonstrate the relationship between k-Markov sequences and operator spreading in a simple case of only SWAP gates. The difference between dynamics can be quantified with cover time, tc, which is the smallest number of circuit layers before every site has hosted σz at least once <ref:2603.182177, The difference between dynamics can be quantified with cover time, tc, which is the smallest number of circuit layers before every site has hosted σz at least once. In the unbiased 0-Markov model, each layer of the circuit is chosen independently and with equal probability <ref:2603.1821778, In the unbiased 0-Markov model, each layer of the circuit is chosen independently and with equal probability. For a 1-Markov process, the cover time will be correlated with the probability of optimal sequences (ASWAPBSWAP or BSWAPASWAP), which is governed entirely by the probability to switch letters, Pswitch = P(AB) = P(BA) <ref:2603.182178, For 1-Markov processes, the cover time will be correlated with the probability of optimal sequences (ASWAPBSWAP or BSWAPASWAP), which is governed entirely by the probability to switch letters, Pswitch = P(AB) = P(BA).

Operator Growth and Correlation Structure

When considering alphabets built from SWAP gates and the set of Clifford gates that can change operator weight, a standard brickwork circuit with gates drawn at random from this set will increase operator weight faster than a standard brickwork circuit with gates drawn at random from the weight-changing (WC) set <ref:2603.182179, A standard brickwork circuit with gates drawn at random from this set will increase operator weight faster thanks to the removal of the weightFIG.. The scrambling time, tscr, is defined as the first layer for which the operator weight satisfies w(l) ≥ N/2 <ref:2603.182179, The scrambling time, tscr, defined as the first layer for which the operator weight satisfies w(l) ≥ N/2. Different k-Markov sequences may generate distinguishable spatial-temporal correlations that are strongest before ⟨tscr⟩ but also persist at very late times <ref:2603.182179, While different k-Markov sequences may give the same mean scrambling time, they generate distinguishable spatial-temporal correlations that are strongest before ⟨tscr⟩, but also persist at very late times.

PSWAP Circuits

Finally, PSWAP circuits are considered where operator weight and complexity can both grow <ref:2603.182179, Finally, we consider a class of circuits where operator weight and complexity can both grow. These PSWAP circuits map a Pauli string into a linear combination of Pauli strings <ref:2603.182179, PSWAP circuits map a Pauli string into a linear combination of Pauli strings. The statistics for m, n in other cases are those of the same-letter run lengths of the k-Markov sequence <ref:2603.182179, The statistics for m, n in other cases are those of the same-letter run lengths of the k-Markov sequence. At k = 1, the sequences generate distributions over a discrete set of PSWAP angles that are independently geometric for odd and even layers <ref:2603.182179, At k = 1, the sequences generate distributions over a discrete set of PSWAP angles that are independently geometric for odd and even layers. At k ≥ 3, the distributions for temporally separated single-letter run lengths start to be correlated <ref:2603.182179, At k ≥ 3, the distributions for temporally separated single-letter run lengths start to be correlated.

Summary

k-Markov sequences of unitary gates can generate families of stochastic circuits that display a variety of structures in operator evolution <ref:2603.182179, Summary - We have demonstrated that k-Markov sequences of unitary gates can be used to generate families of stochastic circuits that display a variety of structures in operator evolution. Within three choices of 2-letter alphabets, the paper demonstrated that many aspects of operator dynamics may be stochastically controlled, including transport dynamics, average spreading or equilibration times, relative weight in distinct symmetry sectors of the operator orbit, and structure in spatial-temporal correlations across the circuit that remain well after network-averaged equilibrium is achieved <ref:2603.182179, "Within three choices of 2-letter alphabets, we demonstrated that many aspects of operator dynamics may be stochastically controlled, including transport dynamics, average spreading or equilibration times, relative weight in distinct symmetry sectors of the operator orbit, and structure in spatial-temporal correlations across the circuit that remain well after network-averaged equilibrium is achieved".

Acknowledgments

The authors thank Zhen Bi, Tommy Chin <ref:2603.182179, Acknowledgments - We thank Zhen Bi, Tommy Chin. The work of Pei-Jun Huang was supported by the Penn State NSF REU through PHY-2349159 <ref:2603.182179, The work of Pei-Jun Huang was supported by the Penn State NSF REU through PHY-2349159. This work was supported by the NSF through PHY-2310662, and with computational resources supported by the NSF through OAC-2201445 <ref:2603.

1. Identify optimal gate sequences for information spreading control: The paper demonstrates that k-Markov processes can be used to control operator transport and that k-Markov sequences provide operationally inexpensive dials to (statistically) control information dynamics without requiring mid-circuit disturbance of the state or real-time adjustment of the algorithm. This allows AI systems to manipulate how information spreads across qubit networks based on controlled stochastic sequences rather than relying solely on random gate choices.

2. Predict and mitigate scrambling time: The work shows that scrambling times can only increase when BWC, ASWAP each act on the even and odd bricklayers, suggesting a method to intentionally increase or decrease scrambling time based on the k-Markov process parameters. This capability can be used in AI to design circuits that either rapidly scramble information (for fast computation) or slow down information spread (for preserving localized properties).

3. Generate structured spatial-temporal correlations: The paper proves that k-Markov sequences generate distinguishable spatial-temporal correlations that are strongest before tscr, but also persist at very late times. An improved AI system could utilize these long-term correlations to create complex memory structures or predictive models where the state of one part of the network influences distant parts over extended time scales.

4. Optimize operator weight dynamics: By controlling the time evolution of the average Hamming weight and how it relates to distinct symmetry sectors, AI can be used to steer the operator's support. This means an AI could design circuits that specifically increase or decrease the complexity (weight) of the quantum state, potentially enabling more efficient resource allocation in quantum computation tasks.