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Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network: Toward quantum soft computing
Hayato Goto
TL;DR
The paper numerically studies quantum adiabatic evolution in a two-spin Ising problem using nonlinear oscillators initialized in the vacuum state. The simulations show that the evolution generates an entangled cat state, with higher fidelity when the nonlinear parameter is increased more slowly.
Problem
The paper examines whether quantum adiabatic evolution can generate an entangled cat state in a two-spin Ising problem.
Method
The Schrödinger equation is solved numerically with a truncated Hilbert space while p increases linearly from zero to 5K.
Results
High fidelities demonstrate that the entangled cat state can be generated, and slower increases of p produce higher fidelity.
Takeaways & Limitations
Quantum adiabatic evolution provides a simple method for deterministic generation of an entangled cat state.
Abstract
from arXiv · showhide
The dynamics of nonlinear systems qualitatively change depending on their parameters, which is called bifurcation. A quantum-mechanical nonlinear oscillator can yield a quantum superposition of two oscillation states, known as a Schrödinger cat state, via quantum adiabatic evolution through its bifurcation point. Here we propose a quantum computer comprising such quantum nonlinear oscillators, instead of quantum bits, to solve hard combinatorial optimization problems. The nonlinear oscillator network finds optimal solutions via quantum adiabatic evolution, where nonlinear terms are increased slowly, in contrast to conventional adiabatic quantum computation or quantum annealing, where quantum fluctuation terms are decreased slowly. As a result of numerical simulations, it is concluded that quantum superposition and quantum fluctuation work effectively to find optimal solutions. It is also notable that the present computer is analogous to neural computers, which are also networks of nonlinear components. Thus, the present scheme will open new possibilities for quantum computation, nonlinear science, and artificial intelligence.
Appendix A: The Wigner function
The appendix defines the Wigner function for oscillator states and describes its phase-space and number-representation forms. These formulas are used to generate the Wigner plots in Figs. 1(e) and 1(f).
- The Wigner function W(x, y) is a quasiprobability distribution for the oscillator quadrature amplitudes x and y.
- Integrating W(x, y) over y gives the probability distribution for x, and integrating over x gives the probability distribution for y.
- The Wigner function is rewritten using displacement and parity operators before being expressed in the number representation.
- Equations (A3) and (A4) were used to calculate the Wigner functions shown in Figs. 1(e) and 1(f).
Appendix B: Calculations of success probabilities and residual energies
The appendix explains how success probabilities and residual energies are calculated from measurements of quadrature-amplitude signs. Wigner-function expressions provide the probabilities of positive and negative outcomes.
- Success probability is the probability that final measurements of quadrature-amplitude signs produce spin configurations corresponding to the solution.
- The residual energy is obtained by taking expectation values using the probabilities of the measured spin configurations.
- The probability of obtaining a positive quadrature amplitude is calculated from the Wigner function using polar coordinates.
- The probability of obtaining a negative quadrature amplitude is calculated analogously from the Wigner function.
Appendix C: Proof of the condition for quantum adiabatic evolution
The appendix proves that positive semidefiniteness of the matrix M is sufficient for the vacuum state |0⟩ to be the ground state when p = 0. It then verifies this condition under the chosen detunings.
- A sufficient condition for |0⟩ to be the ground state of H at p = 0 is that M is positive semidefinite.
- Because H|0⟩ = 0, the proof reduces to showing that H is nonnegative.
- Diagonalizing the Hermitian matrix M expresses H through nonnegative diagonal entries when K and those entries are nonnegative.
- The detunings specified by Eq. (7) make the relevant quadratic form nonnegative, and therefore M is positive semidefinite.
Appendix D: Simulation of the quantum computation for a two-spin Ising problem with a ferromagnetic coupling
The two-spin ferromagnetic Ising simulation targets the two aligned solutions and evaluates generation of their entangled cat state. Numerical evolution shows high fidelity, with slower pumping producing higher fidelity.
- For J1,2 = J2,1 = 1, the ferromagnetic two-spin problem has solutions s1 = s2 = ±1.
- The simulation generates an entangled cat state associated with the aligned two-spin solutions.
- The Schrödinger equation is solved with photon number truncated at 20 for each KPO, starting from |0⟩ while p increases linearly from zero to 5K.
- High fidelity demonstrates entangled-cat-state generation, and increasing p more slowly produces higher fidelity.
- The fidelity F = |⟨ECS(p(t))|ψ(t)⟩|2 is plotted for computation times of 500/K and 200/K.
Appendix E: Approximate solution by the classical model
The classical model relaxes the Ising problem to a continuous optimization whose solution is the eigenvector of M with the smallest eigenvalue. Near the first bifurcation point, the model recovers this eigenvector and thereby obtains an approximate Ising solution.
- The relaxation replaces each Ising spin s_i with a continuous variable ζ_i, then recovers an approximate spin solution from sign(ζ_i).
- The relaxation minimizes an energy under a constraint, and its solution is the eigenvector of M corresponding to the smallest eigenvalue.Because M is positive semidefinite, this relaxation solution also provides a lower bound for the Ising energy.
- Near the first bifurcation point, small quadrature amplitudes justify neglecting nonlinear terms when deriving the fixed-point condition.
- At the bifurcation point, p equals the smallest eigenvalue of M, and the nontrivial classical amplitudes form the corresponding eigenvector.
- Thus, the classical model can find the relaxation solution, which may explain its high probability of finding optimal solutions.