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arXiv:2410.11534

Supersymmetry-Driven Quantum Gate Design Based on Feynman Path Integral and TPCP Map Optimization

Harish Parthasarathy, Monika Aggarwal, Kumar Gautam

Published Oct 15, 2024• 0 citations
arXivHTMLPDF

Abstract

We use supersymmetry to enlarge the dimension of the Hilbert space on which the unitary evolution of the state of the quantum fields acts. We discuss how to control the unitary evolution or TPCP maps generated by the quantum evolution of the fields by controlling the vacuum expectations of other fields in the theory. This amounts to breaking supersymmetry using control vacuum expectation values of the other fields. The evolution of the wave functional or TPCP maps obtained by tracing out over other fields is based on the Feynman path integral formula for the fields. By using the methods of quantum stochastic filtering, we estimate the evolving state of the fields from non-demolition noise measurements and then design a family of TPCP maps evolving in time whose outputs match the estimated evolving state. In this way, we are able to simulate the evolution of the state of the quantum noisy fields. Direct matching of the designed TPCP map to output the evolving system state is not possible since there is no way by which we can determine the exact evolving state, we can only estimate it using non-demolition measurements. The family of designed TPCP maps can be based on using a simulated master equation with unknown parameters incorporated into the Hamiltonian and the other Lindblad operators, chosen so as to match the state outputted by the quantum filter.

DOI
10.48550/arxiv.2410.11534

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We use supersymmetry to enlarge the dimension of the Hilbert space on which the unitary evolution of the state of the quantum fields acts. We discuss how to control the unitary evolution or TPCP maps generated by the quantum evolution of the fields by controlling the vacuum expectations of other fields in the theory. This amounts to breaking supersymmetry using control vacuum expectation values of the other fields. The evolution of the wave functional or TPCP maps obtained by tracing out over other fields is based on the Feynman path integral formula for the fields. By using the methods of quantum stochastic filtering, we estimate the evolving state of the fields from non-demolition noise measurements and then design a family of TPCP maps evolving in time whose outputs match the estimated evolving state. In this way, we are able to simulate the evolution of the state of the quantum noisy fields. Direct matching of the designed TPCP map to output the evolving system state is not possible since there is no way by which we can determine the exact evolving state, we can only estimate it using non-demolition measurements. The family of designed TPCP maps can be based on using a simulated master equation with unknown parameters incorporated into the Hamiltonian and the other Lindblad operators, chosen so as to match the state outputted by the quantum filter.
We use supersymmetry to enlarge the dimension of the Hilbert space on which the unitary evolution of the state of the quantum fields acts. We discuss how to control the unitary evolution or TPCP maps generated by the quantum evolution of the fields by controlling the vacuum expectations of other fields in the theory. This amounts to breaking supersymmetry using controlled vacuum expectation values of the other fields. The evolution of the wave functional or TPCP maps obtained by tracing out over other fields is based on the Feynman path integral formula for the fields. By using the methods of quantum stochastic filtering, we estimate the evolving state of the fields from non-demolition noise measurements and then design a family of TPCP maps evolving in time whose outputs match the estimated evolving state. In this way, we are able to simulate the evolution of the state of the quantum noisy fields. Direct matching of the designed TPCP map to output the evolving system state is not possible since there is no way by which we can determine the exact evolving state; we can only estimate it using non-demolition measurements. The family of designed TPCP maps can be based on using a simulated master equation with unknown parameters incorporated into the Hamiltonian and the other Lindblad operators, chosen so as to match the state outputted by the quantum filter.
Belavkin first constructed non-demolition measurements for such systems using which one could construct the conditional expectation of a Heisenberg observable evolving in the HPS-qsde framework and that this conditional expectation could be computed on a real time basis leading to the Belavkin filter that generalizes the classical Kushner-Kallianpur filter in stochastic filtering theory. Physical meaning to the filtering results in such coherent states can be obtained only after averaging the results by means of a weight function dependent upon the Grassmannian parameters using the Berezin integral. This idea also plays a fundamental role in the theory of quantum antennas wherein the electromagnetic field produced by the current density generated by electrons and positrons in the Dirac picture is a quadratic combination of the Fermionic creation and annihilation operators and hence by applying the retarded potential formula, the electromagnetic field radiated out by such a quantum antenna is also a quadratic functional of the Fermionic creation and annihilation operators.
In this paper, we first derive the discrete time recursive stochastic filtering equations for a classical Markov process when noisy measurements of the process are made with the measurement noise at each discrete sample forms an iid Gaussian or non-Gaussian process. The filter involves deriving general formulae for the conditional mean process given measurements upto that time and also simultaneously recursive formulae for the conditional moments of the estimation error process. In the next section, we build upon the method of John Gough etal, to construct a supersymmetric quantum stochastic filter, ie, the supersymmetric Belavkin filter. The construction of supersymmetric noise in the Hudson Parthasarathy noisy Schrodinger equation is based on constructing Fermionic noise by applying a twist to Bosonic noise, ie, dJ=(−1)ΛdA,dJ∗=(−1)ΛdA∗dJ=(-1)^{\Lambda}dA, dJ^*=(-1)^{\Lambda}dA^*dJ=(−1)ΛdA,dJ∗=(−1)ΛdA∗. Such a Fermionic noise has memory unlike the Bosonic noise and satisfies the CAR. We prove that when a superposition of Bosonic quantum Brownian motion and Bosonic counting process along with Fermionic counting process is measured through the Hudson-Parthasarathy noisy Schrodinger system with both Bosonic and supersymmetric noise, then it satisfies the non-demolition property and hence conditional expectations can be defined. The construction of the quantum filter here involves taking into consideration all the positive integer powers of the output measurement noise differential and we give an algorithm for computing these powers based on the quantum Ito formula. The method for deriving a countably infinite number of linear equations for the filter coefficients is based on the reference probability method of Gough et.al. The method of constructing supersymmetric noise for driving the HPS equation is based on the work of Timothy Eyre. The main result of this paper is to derive an infinite system of linear equations for the quantum filter coefficients thereby yielding a real time implementable filter for estimating system observables or equivalently, the system mixed state from the most general form of non-demolition measurements comprising Bosonic continuous and counting noise plus Fermionic counting noise.
In this paper, we first derive the discrete time recursive stochastic filtering equations for a classical Markov process when noisy measurements of the process are made with the measurement noise at each discrete sample forms an iid Gaussian or non-Gaussian process. The filter involves deriving general formulae for the conditional mean process given measurements upto that time and also simultaneously recursive formulae for the conditional moments of the estimation error process. In the next section, we build upon the method of John Gough etal, to construct a supersymmetric quantum stochastic filter, ie, the supersymmetric Belavkin filter. The construction of supersymmetric noise in the Hudson Parthasarathy noisy Schrodinger equation is based on constructing Fermionic noise by applying a twist to Bosonic noise, ie, dJ=(−1)ΛdA,dJ∗=(−1)ΛdA∗dJ=(-1)^{\Lambda}dA, dJ^*=(-1)^{\Lambda}dA^*dJ=(−1)ΛdA,dJ∗=(−1)ΛdA∗. Such a Fermionic noise has memory unlike the Bosonic noise and satisfies the CAR. We prove that when a superposition of Bosonic quantum Brownian motion and Bosonic counting process along with Fermionic counting process is measured through the Hudson-Parthasarathy noisy Schrodinger system with both Bosonic and supersymmetric noise, then it satisfies the non-demolition property and hence conditional expectations can be defined. The construction of the quantum filter here involves taking into consideration all the positive integer powers of the output measurement noise differential and we give an algorithm for computing these powers based on the quantum Ito formula. The method for deriving a countably infinite number of linear equations for the filter coefficients is based on the reference probability method of Gough et.al. The method of constructing supersymmetric noise for driving the HPS equation is based on the work of Timothy Eyre. The main result of this paper is to derive an infinite system of linear equations for the quantum filter coefficients thereby yielding a real time implementable filter for estimating system observables or equivalently, the system mixed state from the most general form of non-demolition measurements comprising Bosonic continuous and counting noise plus Fermionic counting noise.
We consider the problem of optimal estimation of a state of a quantum system interacting with the Bose field being in a mixture of the vacuum and coherent state.The system and electromagnetic field are described by making use of quantum stochastic unitary evolution.We derive stochastic master equation using Gardiner and Collet's input-output theory and the concept of cascade systems.To find the stochastic evolution of the system conditioned on the result of the quadrature measurement of the output field, we extend the compound system by an ancilla driven by the vacuum and generating the desired non-classical state of light.
A unitary evolution in time may be treated as a curve in the manifold of the special unitary group. The length of such a curve can be related to the energetic cost of the associated computation, meaning a geodesic curve identifies an energetically optimal path. In this work, we employ sub-Riemannian geometry on the manifold of the unitary group to obtain optimally designed Hamiltonians for generating single-qubit gates in an environment with the presence of dephasing noise as well as a two-qubit gate under a time-constant crosstalk interaction. The resulting geodesic equation involves knowing the initial conditions of the parameters that cannot be obtained analytically. We then introduce a random sampling method combined with a minimization function and a cost function to find initial conditions that lead to optimal control fields. We also compare the optimized control fields obtained from the solutions of the geodesic equation with those extracted from the well-known Krotov method. Both approaches provide high-fidelity values for the desired quantum gate implementation, but the geodesic method has the advantage of minimizing the required energy to execute the same task. These findings bring insights for the design of more efficient fields in the arsenal of optimal control theory.