Simple Proof of Equivalence between Adiabatic Quantum Computation and the Circuit Model

Abstract

We prove the equivalence between adiabatic quantum computation and quantum computation in the circuit model. An explicit adiabatic computation procedure is given that generates a ground state from which the answer can be extracted. The amount of time needed is evaluated by computing the gap. We show that the procedure is computationally efficient.

Noise-enhanced performance of adiabatic quantum computing by lifting degeneracies

“…3), which corresponds to a system of 4 physical qubits. Following the procedure described in [7] we can then write;…”

“…As mentioned previously AQC has been shown to be polynomially equivalent to the circuit model of quantum computing.The 'ground state quantum computing' (GSQC) formalism, described in [7], offers the most practical method of constructing a H 0 that encodes an arbitrary M qubit, N step quantum circuit. In the GSQC formalism, each of the M qubits in the circuit is viewed as a single electron that can occupy the states in an array of 2×(N +1) quantum dots; where the rows in the array represents either the |0 or |1 states of the qubit.…”

“…This scheme is believed to have a number of advantages over the "standard" approach, namely, the precise time-dependent control of individual qubits is no longer required, and it benefits from an inherent robustness against some of the effects of decoherence by remaining in the instantaneous ground state at all times [2][3][4]. Crucially, AQC has been shown to be polynomially equivalent, under certain conditions, to the standard gate model of quantum computing [6,7]. The effects of noise on AQC are generally considered to be detrimental but manageable [2][3][4].…”

Quantum gates with controlled adiabatic evolutions

“…and constructing the slightly more complicated threelocal Hamiltonian 2 :Ĥ (t) = |1,n ⊥ 1,n ⊥ | ⊗Ĥ φ (t) (9) + (|0,n 0,n| + |0,n ⊥ 0,n ⊥ | + |1,n 1,n|) ⊗Ĥ 0 (t) , whereĤ 0 (t) andĤ φ (t) are as previously defined. In the limit of θ f → π, the end state in this case will be, the controlled-rotated state |ψ c. rot.…”

“…The aforementioned experimental studies, as well as other theoretical work such as the theorem of polynomial equivalence between AQC and the predominant gate model (GM) paradigm of quantum computing [8,9], provide ample motivation for determining the computational capabilities of AQC and its precise relations with other quantum computing paradigms, specifically GM. Demonstrating that algorithms such as Shor's integer factorization [1] are implementable as efficiently on a quantum adiabatic computer would undoubtedly have many practical as well as theoretical consequences that would resonate well beyond Quantum Computing.…”