Kerr parametric oscillators (KPOs) have attracted increasing attention in terms of their application to quantum information processing and quantum simulations. The state preparation and measurement of KPOs are typical requirements when they are used as qubits. The methods previously proposed for state preparations of KPOs utilize modulation of a pump field or an auxiliary drive field. We study the stochastic state preparation of a KPO based on homodyne detection, which does not require modulation of a pump field nor an auxiliary drive field, and thus can exclude unwanted effects of possible imperfection in control of these fields. We quantitatively show that the detection data, if averaged over a proper time to decrease the effect of measurement noise, has a strong correlation with the state of the KPO, and therefore can be used to estimate the state of the KPO (stochastic state preparation). We examine the success probability of the state estimation taking into account the effect of the measurement noise and bit flips. Moreover, the proper range of the averaging time to realize a high success probability is obtained by developing a binomial-coherent-state model, which describes the stochastic dynamics of the KPO under homodyne detection.Kerr parametric oscillators (KPOs) have attracted increasing attention in terms of their application to quantum information processing and quantum simulations. The state preparation and measurement of KPOs are typical requirements when they are used as qubits. The methods previously proposed for state preparations of KPOs utilize modulation of a pump field or an auxiliary drive field. We study the stochastic state preparation of a KPO based on homodyne detection, which does not require modulation of a pump field nor an auxiliary drive field, and thus can exclude unwanted effects of possible imperfection in control of these fields. We quantitatively show that the detection data, if averaged over a proper time to decrease the effect of measurement noise, has a strong correlation with the state of the KPO, and therefore can be used to estimate the state of the KPO (stochastic state preparation). We examine the success probability of the state estimation taking into account the effect of the measurement noise and bit flips. Moreover, the proper range of the averaging time to realize a high success probability is obtained by developing a binomial-coherent-state model, which describes the stochastic dynamics of the KPO under homodyne detection.

arXiv:2208.04542Aug 9, 20220 citations

A Constructive Unitary Model of Quantum Measurement: Born-Consistent Outcomes and Stable Records

fei ding, George S. Zhou

We construct a finite quantum measurement model using a quantum-nondemolition (QND) qubit coupled to a two-photon-driven Kerr parametric oscillator (KPO) with continuously monitored homodyne output. A unitary dilation of the qubit, pointer, and field accounts for decoherence, output noise, and conditioning; the stochastic master equation is a filter for the observed record, not an objective-collapse law. Exact QND commutation makes the conditioned source-sector weight a bounded martingale. Under asymptotic output identifiability, it becomes extremal almost surely, with terminal hitting probabilities equal to the initial Born sector weights. This establishes instrument-level Born consistency, not a derivation of the Hilbert-space probability rule from dynamics alone. The nonlinear KPO amplifies sector-dependent biases and registers information in metastable pointer basins. Pointer separation scales as $\sqrt{N}$ and the effective semiclassical noise scale as $N^{-1/2}$ . Full quantum trajectories show synchronization of conditional extremalization and pointer locking. Over the simulated finite-size range, wrong registration and later record corruption decrease approximately exponentially with distinct rates. Reduced first-passage, Liouvillian-metastability, and Freidlin-Wentzell minimum-action calculations independently support the corruption exponent to within a few percent. Thus conditional extremalization, single-basin registration, and metastable storage coexist within standard unitary-background quantum mechanics, without deleting unconditioned global components or adding a fundamental stochastic selector. Classical actuality remains a separate record-level ontological claim. The stored KPO record is a local building block for the companion Framework, whose broader claims require persistent co-recordable records in one commutative record algebra.We construct a finite quantum measurement model using a quantum-nondemolition (QND) qubit coupled to a two-photon-driven Kerr parametric oscillator (KPO) with continuously monitored homodyne output. A unitary dilation of the qubit, pointer, and field accounts for decoherence, output noise, and conditioning; the stochastic master equation is a filter for the observed record, not an objective-collapse law. Exact QND commutation makes the conditioned source-sector weight a bounded martingale. Under asymptotic output identifiability, it becomes extremal almost surely, with terminal hitting probabilities equal to the initial Born sector weights. This establishes instrument-level Born consistency, not a derivation of the Hilbert-space probability rule from dynamics alone. The nonlinear KPO amplifies sector-dependent biases and registers information in metastable pointer basins. Pointer separation scales as $\sqrt{N}$ and the effective semiclassical noise scale as $N^{-1/2}$. Full quantum trajectories show synchronization of conditional extremalization and pointer locking. Over the simulated finite-size range, wrong registration and later record corruption decrease approximately exponentially with distinct rates. Reduced first-passage, Liouvillian-metastability, and Freidlin-Wentzell minimum-action calculations independently support the corruption exponent to within a few percent. Thus conditional extremalization, single-basin registration, and metastable storage coexist within standard unitary-background quantum mechanics, without deleting unconditioned global components or adding a fundamental stochastic selector. Classical actuality remains a separate record-level ontological claim. The stored KPO record is a local building block for the companion Framework, whose broader claims require persistent co-recordable records in one commutative record algebra.

Sep 7, 20260 citations

Quantum Gate for a Kerr Nonlinear Parametric Oscillator Using Effective Excited States

Taro Kanao, Shumpei Masuda, Shiro Kawabata +1 more

A Kerr nonlinear parametric oscillator (KPO) can stabilize a quantum superposition of two coherent states with opposite phases, which can be used as a qubit. In a universal gate set for quantum computation with KPOs, an ${R}_{x}$ gate, which interchanges the two coherent states, is relatively hard to perform owing to the stability of the two states. We propose a method for a high-fidelity ${R}_{x}$ gate by exciting the KPO outside the qubit space with parity-selective transitions, which can be implemented by only adding a driving field. In this method, the utilization of higher effective excited states leads to a faster ${R}_{x}$ gate, rather than states near the qubit space. The proposed method can realize a continuous ${R}_{x}$ gate and thus is expected to be useful for, e.g., recently proposed variational quantum algorithms.A Kerr nonlinear parametric oscillator (KPO) can stabilize a quantum superposition of two coherent states with opposite phases, which can be used as a qubit. In a universal gate set for quantum computation with KPOs, an ${R}_{x}$ gate, which interchanges the two coherent states, is relatively hard to perform owing to the stability of the two states. We propose a method for a high-fidelity ${R}_{x}$ gate by exciting the KPO outside the qubit space with parity-selective transitions, which can be implemented by only adding a driving field. In this method, the utilization of higher effective excited states leads to a faster ${R}_{x}$ gate, rather than states near the qubit space. The proposed method can realize a continuous ${R}_{x}$ gate and thus is expected to be useful for, e.g., recently proposed variational quantum algorithms.

arXiv:2108.03091Physical Review AppliedJul 8, 202232 citations

Three Strongly Coupled Kerr Parametric Oscillators Forming a Boltzmann Machine

Gabriel Margiani, Orjan Ameye, Oded Zilberberg +1 more

Coupled Kerr parametric oscillators (KPOs) are a promising resource for classical and quantum analog computation, for example to find the ground state of Ising Hamiltonians. Yet, the state space of strongly coupled KPO networks is very involved. As such, their phase diagram sometimes features either too few or too many states, including some that cannot be mapped to Ising spin configurations. This complexity makes it challenging to find and meet the conditions under which an analog optimization algorithm can be successful. Here, we demonstrate how to use three strongly coupled KPOs as a simulator for an Ising Hamiltonian, and estimate its ground state using a Boltzmann sampling measurement. While fully classical, our Letter is directly relevant for quantum systems operating on coherent states.Coupled Kerr parametric oscillators (KPOs) are a promising resource for classical and quantum analog computation, for example to find the ground state of Ising Hamiltonians. Yet, the state space of strongly coupled KPO networks is very involved. As such, their phase diagram sometimes features either too few or too many states, including some that cannot be mapped to Ising spin configurations. This complexity makes it challenging to find and meet the conditions under which an analog optimization algorithm can be successful. Here, we demonstrate how to use three strongly coupled KPOs as a simulator for an Ising Hamiltonian, and estimate its ground state using a Boltzmann sampling measurement. While fully classical, our Letter is directly relevant for quantum systems operating on coherent states.

arXiv:2504.04254Physical Review LettersJul 29, 20253 citations

Quantum State Tomography for Kerr Parametric Oscillators

Yuta Suzuki, Shiro Kawabata, Tsuyoshi Yamamoto +1 more

Kerr parametric oscillators (KPOs) implemented in the circuit quantum electrodynamic architecture can operate as qubits. Their applications to quantum annealing and universal quantum computation have been studied intensively. For these applications, the readout of the state of KPOs is of practical importance. We develop a scheme of state tomography for KPOs with reflection measurement. Although it is known that the reflection coefficient depends on the state of the KPO, it is unclear whether tomography of a qubit encoded into a KPO can be performed in a practical way mitigating decoherence during the measurement, and how accurate it is. We show that the reflection coefficient has a one-to-one correspondence with a diagonal element of the density matrix of the qubit when a probe frequency is properly chosen and an additional single-photon drive is introduced. Thus, our scheme offers a novel way to read out the qubit along an axis of the Bloch sphere, and therefore the reflection measurement and single-qubit gates can constitute state tomography.Kerr parametric oscillators (KPOs) implemented in the circuit quantum electrodynamic architecture can operate as qubits. Their applications to quantum annealing and universal quantum computation have been studied intensively. For these applications, the readout of the state of KPOs is of practical importance. We develop a scheme of state tomography for KPOs with reflection measurement. Although it is known that the reflection coefficient depends on the state of the KPO, it is unclear whether tomography of a qubit encoded into a KPO can be performed in a practical way mitigating decoherence during the measurement, and how accurate it is. We show that the reflection coefficient has a one-to-one correspondence with a diagonal element of the density matrix of the qubit when a probe frequency is properly chosen and an additional single-photon drive is introduced. Thus, our scheme offers a novel way to read out the qubit along an axis of the Bloch sphere, and therefore the reflection measurement and single-qubit gates can constitute state tomography.

arXiv:2212.14627Physical Review AppliedSep 14, 20237 citations

State Estimation and Control for Stochastic Quantum Dynamics With Homodyne Measurement: Stabilizing Qubits Under Uncertainty

Nahid Binandeh Dehaghani, A. Pedro Aguiar, Rafał Wiśniewski

This paper introduces a Lyapunov-based control strategy alongside two filtering methods for controlling and estimating the evolution of coherence vector elements from sequential homodyne measurements. The methods include traditional quantum filtering and a novel extended Kalman filter, which explicitly addresses the dynamics of a stochastic master equation with correlated noise, thereby ensuring the quantum properties of the estimated state variable by design. We also explore scenarios where the system’s Hamiltonian is unknown, demonstrating that both filters exhibit reduced performance with increased estimation errors. To address this, we propose a multiple model estimation scheme applicable to either filter. The estimated density operator is then controlled using the proposed switching-based Lyapunov scheme, which guarantees noise-to-state practical stability in probability. We demonstrate the effectiveness of our approach in stabilizing a qubit coupled to a leaky cavity under homodyne detection, even with uncertainty in the resonance frequency.This paper introduces a Lyapunov-based control strategy alongside two filtering methods for controlling and estimating the evolution of coherence vector elements from sequential homodyne measurements. The methods include traditional quantum filtering and a novel extended Kalman filter, which explicitly addresses the dynamics of a stochastic master equation with correlated noise, thereby ensuring the quantum properties of the estimated state variable by design. We also explore scenarios where the system’s Hamiltonian is unknown, demonstrating that both filters exhibit reduced performance with increased estimation errors. To address this, we propose a multiple model estimation scheme applicable to either filter. The estimated density operator is then controlled using the proposed switching-based Lyapunov scheme, which guarantees noise-to-state practical stability in probability. We demonstrate the effectiveness of our approach in stabilizing a qubit coupled to a leaky cavity under homodyne detection, even with uncertainty in the resonance frequency.

arXiv:2403.07021IEEE AccessJan 1, 20242 citations

\sqrt{N}

and the effective semiclassical noise scale as

N^{-1/2}

. Full quantum trajectories show synchronization of conditional extremalization and pointer locking. Over the simulated finite-size range, wrong registration and later record corruption decrease approximately exponentially with distinct rates. Reduced first-passage, Liouvillian-metastability, and Freidlin-Wentzell minimum-action calculations independently support the corruption exponent to within a few percent. Thus conditional extremalization, single-basin registration, and metastable storage coexist within standard unitary-background quantum mechanics, without deleting unconditioned global components or adding a fundamental stochastic selector. Classical actuality remains a separate record-level ontological claim. The stored KPO record is a local building block for the companion Framework, whose broader claims require persistent co-recordable records in one commutative record algebra.

{R}_{x}

gate, which interchanges the two coherent states, is relatively hard to perform owing to the stability of the two states. We propose a method for a high-fidelity

{R}_{x}

gate by exciting the KPO outside the qubit space with parity-selective transitions, which can be implemented by only adding a driving field. In this method, the utilization of higher effective excited states leads to a faster

{R}_{x}

gate, rather than states near the qubit space. The proposed method can realize a continuous

{R}_{x}

gate and thus is expected to be useful for, e.g., recently proposed variational quantum algorithms.