Physics and computation: An insight from non-Hermitian quantum computing

TL;DR

It is shown that NQC is extraordinarily powerful, capable of solving not only all NP problems but also all problems within the complexity class P in polynomial time.

Abstract

We elucidate the profound connection between physics and computation by proposing and examining the model of the non-Hermitian quantum computer (NQC). In addition to conventional quantum gates such as the Hadamard, phase, and CNOT gates, this model incorporates a non-unitary quantum gate $G$. We show that NQC is extraordinarily powerful, capable of solving not only all NP problems but also all problems within the complexity class $\text{P}^{\sharp\text{P}}$ in polynomial time. We investigate two physical schemes for implementing the non-unitary gate $G$ and find that the remarkable computational power of NQC originates from the exponentially large physical resources required.

Figures (5)

• Figure 1: (a) Two-bit logical controlled-$G$ (C-$G$) gate; (b) A method to implement the C-$G$ gate using CNOT and $G$ gates. In this scheme, the parameter $g'$ for the C-$G$ gate in (a) is related to the parameter $g$ of the $G$ gate in (b) as $g'=g^2$.
• Figure 2: Circuit of an NQC algorithm for solving the MIS problem, which is NP-hard. The big box represents the oracle that implements the operator (\ref{['misOracle']}). The state $|0_n\rangle$ with subscript $n$ stands for the state of the qubit that may undergo non-unitary transformation. The gate $H$ represents the Hadamard gate. The gate $G$ and $G^r$ is for the gate shown in Eqs. (\ref{['Phi']}) and (\ref{['Phir']}). As explained in the text, $r$ is proportional to $n$. The controlled gates are $C-G^{r}$ to correctly implement the amplitude scaling in Eq. (11).
• Figure 3: (a) Traditional qubits are realized using a single-particle double-well system, while qubits capable of non-unitary operations are realized using a multi-particle double-well system. Using Feshbach resonance, we achieve zero inter-particle interactions in the multi-particle system, while particles in the multi-particle system interact with those in the single-particle system. (b) The CZ gate is implemented by bringing one well of the single-particle system close to one well of the multi-particle system. The potential barrier between the two double-well systems is sufficiently high, such that particles in each double-well system cannot tunnel into the other system. This CZ gate, as shown in the Appendix A, can be directly interpreted as a CNOT gate in the computational basis.
• Figure 4: Using the CZ and Hadamard gates $H$ to realize the CNOT gate.
• Figure 5: The degenerate (exceptional points) circle $x^2+y^2=s^2$ for the Hamiltonian (\ref{['gene-H']}) and a suitable loop in the $x-y$ parameter space for adiabatic control to generate a purely imaginary geometric phase to realize the logic gate $G$ and $G^r$. By manipulating parameters $(x,y)$ along the loop for one full revolution, $G$ is implemented; for $r$ revolutions, $G^r$ is achieved.