We experimentally implement the Sz.-Nagy dilation algorithm to simulate open quantum dynamics on a nuclear magnetic resonance quantum processor. The Sz.-Nagy algorithm enables the simulation of the dynamics of an $n$ -qubit system using $n+1$ qubits. We experimentally simulate the action of three nonunitary processes, namely, a phase damping channel acting independently on two qubits, a two-qubit correlated amplitude damping channel, and a magnetic-field-gradient pulse acting on an ensemble of two coupled nuclear spin- $\frac{1}{2}$ particles. To evaluate the quality of the experimentally simulated quantum process, we perform convex-optimization-based full quantum process tomography to reconstruct the quantum process from the experimental data and compare it with the target quantum process to be simulated.We experimentally implement the Sz.-Nagy dilation algorithm to simulate open quantum dynamics on a nuclear magnetic resonance quantum processor. The Sz.-Nagy algorithm enables the simulation of the dynamics of an $n$-qubit system using $n+1$ qubits. We experimentally simulate the action of three nonunitary processes, namely, a phase damping channel acting independently on two qubits, a two-qubit correlated amplitude damping channel, and a magnetic-field-gradient pulse acting on an ensemble of two coupled nuclear spin-$\frac{1}{2}$ particles. To evaluate the quality of the experimentally simulated quantum process, we perform convex-optimization-based full quantum process tomography to reconstruct the quantum process from the experimental data and compare it with the target quantum process to be simulated.

arXiv:2201.07687Physical Review AAug 22, 202213 citations

POVMs and the Two Theorems of Naimark and Sz.-Nagy

James D. Malley, Anthony R. Fletcher

In 1940 Naimark showed that if a set of quantum observables are positive semi-definite and sum to the identity then, on a larger space, they have a joint resolution as commuting projectors. In 1955 Sz.-Nagy showed that any set of observables could be so resolved, with the resolution respecting all linear sums. Crucially, both resolutions return the correct Born probabilities for the original observables. Here, an alternative proof of the Sz.-Nagy result is given using elementary inner product spaces. A version of the resolution is then shown to respect all products of observables on the base space. Practical and theoretical consequences are indicated. For example, quantum statistical inference problems that involve any algebraic functionals can now be studied using classical statistical methods over commuting observables. The estimation of quantum states is a problem of this type. Further, as theoretical objects, classical and quantum systems are now distinguished by only more or less degrees of freedom.In 1940 Naimark showed that if a set of quantum observables are positive semi-definite and sum to the identity then, on a larger space, they have a joint resolution as commuting projectors. In 1955 Sz.-Nagy showed that any set of observables could be so resolved, with the resolution respecting all linear sums. Crucially, both resolutions return the correct Born probabilities for the original observables. Here, an alternative proof of the Sz.-Nagy result is given using elementary inner product spaces. A version of the resolution is then shown to respect all products of observables on the base space. Practical and theoretical consequences are indicated. For example, quantum statistical inference problems that involve any algebraic functionals can now be studied using classical statistical methods over commuting observables. The estimation of quantum states is a problem of this type. Further, as theoretical objects, classical and quantum systems are now distinguished by only more or less degrees of freedom.

arXiv:1509.06999AxiomsSep 1, 20150 citations

Divide-and-Conquer Simulation of Open Quantum Systems

Thiago Melo D. Azevedo, Almeida, Caio, Pedro Linck Maciel +2 more

One of the promises of quantum computing is to simulate physical systems efficiently. However, the simulation of open quantum systems - where interactions with the environment play a crucial role - remains challenging for quantum computing, as it is impossible to implement deterministically non-unitary operators on a quantum computer without auxiliary qubits. The Stinespring dilation can simulate an open dynamic but requires a high circuit depth, which is impractical for NISQ devices. An alternative approach is parallel probabilistic block-encoding methods, such as the Sz.-Nagy and Singular Value Decomposition dilations. These methods result in shallower circuits but are hybrid methods, and we do not simulate the quantum dynamic on the quantum computer. In this work, we describe a divide-and-conquer strategy for preparing mixed states to combine the output of each Kraus operator dilation and obtain the complete dynamic on quantum hardware with a lower circuit depth. The work also introduces a balanced strategy that groups the original Kraus operators into an expanded operator, leading to a trade-off between circuit depth, CNOT count, and number of qubits. We perform a computational analysis to demonstrate the advantages of the new method and present a proof-of-concept simulation of the Fenna-Matthews-Olson dynamic on current quantum hardware.One of the promises of quantum computing is to simulate physical systems efficiently. However, the simulation of open quantum systems - where interactions with the environment play a crucial role - remains challenging for quantum computing, as it is impossible to implement deterministically non-unitary operators on a quantum computer without auxiliary qubits. The Stinespring dilation can simulate an open dynamic but requires a high circuit depth, which is impractical for NISQ devices. An alternative approach is parallel probabilistic block-encoding methods, such as the Sz.-Nagy and Singular Value Decomposition dilations. These methods result in shallower circuits but are hybrid methods, and we do not simulate the quantum dynamic on the quantum computer. In this work, we describe a divide-and-conquer strategy for preparing mixed states to combine the output of each Kraus operator dilation and obtain the complete dynamic on quantum hardware with a lower circuit depth. The work also introduces a balanced strategy that groups the original Kraus operators into an expanded operator, leading to a trade-off between circuit depth, CNOT count, and number of qubits. We perform a computational analysis to demonstrate the advantages of the new method and present a proof-of-concept simulation of the Fenna-Matthews-Olson dynamic on current quantum hardware.

arXiv:2505.01623May 2, 20250 citations

n

-qubit system using

n+1

qubits. We experimentally simulate the action of three nonunitary processes, namely, a phase damping channel acting independently on two qubits, a two-qubit correlated amplitude damping channel, and a magnetic-field-gradient pulse acting on an ensemble of two coupled nuclear spin-

\frac{1}{2}

particles. To evaluate the quality of the experimentally simulated quantum process, we perform convex-optimization-based full quantum process tomography to reconstruct the quantum process from the experimental data and compare it with the target quantum process to be simulated.