The Imminent Failure of ‘Adiabatic’ Quantum Computing & Quantum Annealing!

What is ‘Adiabatic’ Quantum Computing?

Adiabatic quantum computation (AQC) is a form of quantum computing which relies on the adiabatic theorem to do calculations and is closely related to, and may be regarded as a subclass of, quantum annealing.

First, a (potentially complicated) Hamiltonian is found whose ground state describes the solution to the problem of interest. Next, a system with a simple Hamiltonian is prepared and initialized to the ground state. Finally, the simple Hamiltonian is adiabatically evolved to the desired complicated Hamiltonian. By the adiabatic theorem, the system remains in the ground state, so at the end the state of the system describes the solution to the problem. Adiabatic quantum computing has been shown to be polynomially equivalent to conventional quantum computing in the circuit model.

The time complexity for an adiabatic algorithm is the time taken to complete the adiabatic evolution which is dependent on the gap in the energy eigenvalues (spectral gap) of the Hamiltonian. Specifically, if the system is to be kept in the ground state, the energy gap between the ground state and the first excited state of H(t) provides an upper bound on the rate at which the Hamiltonian can be evolved at time t. When the spectral gap is small, the Hamiltonian has to be evolved slowly.

What is Quantum Annealing?

Quantum annealing (QA) is a metaheuristic for finding the global minimum of a given objective function over a given set of candidate solutions (candidate states), by a process using quantum fluctuations. Quantum annealing is used mainly for problems where the search space is discrete (combinatorial optimization problems) with many local minima; such as finding the ground state of a spin glass. It was formulated in its present form by T. Kadowaki and H. Nishimori in “Quantum annealing in the transverse Ising model” though a proposal in a different form had been made by A. B. Finilla, M. A. Gomez, C. Sebenik and J. D. Doll, in “Quantum annealing: A new method for minimizing multidimensional functions”.

Quantum annealing starts from a quantum-mechanical superposition of all possible states (candidate states) with equal weights. Then the system evolves following the time-dependent Schrödinger equation, a natural quantum-mechanical evolution of physical systems. The amplitudes of all candidate states keep changing, realizing a quantum parallelism, according to the time-dependent strength of the transverse field, which causes quantum tunneling between states. If the rate of change of the transverse field is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian (also see adiabatic quantum computation). If the rate of change of the transverse field is accelerated, the system may leave the ground state temporarily but produce a higher likelihood of concluding in the ground state of the final problem Hamiltonian, i.e., adiabatic quantum computation. The transverse field is finally switched off, and the system is expected to have reached the ground state of the classical Ising model that corresponds to the solution to the original optimization problem.

What kind of Problems are Vendors with Quantum Annealing Technologies Trying to Solve?

They are basically trying to implement QUBO — Quantum Unconstrained Binary Optimization formulations. And Quantum Sampling formulations. And perhaps some formulations related to Quantum Machine Learning by sampling Machine Learning Parameters from the Search Space.

So What is The Problem? Why is ‘All’ This a Huge Failure?

The time taken for the Quantum Adiabatic/Annealing Algorithm is basically of the order of 1/g² where g is the spectral energy gap.

Even for slightly complex problems with 20–30 variables the energy gap g closes (i.e. diminishes) exponentially fast. e.g. g will goto 0.00000000000000000000000000001 put that in the formula above and you will get the execution time for the Quantum Algorithm.

Which means …

Firstly, the algorithm will always jump the energy gap and destroy the solution.

Secondly, the algorithm will take exponentially long time to solve problems and even small problems with 20–30 variables are intractable.

Scott Aaronson has explained and proven this very elaborately in the past.

Conclusion

Adiabatic Quantum Computing and Quantum Annealing Computers are a Proven Failure due to the underlying basis of the way they work.

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