Quantum chaos $\&$ quantum computers

In [107] we study a generic model of quantum computer, composed of many qubits coupled by short-range interaction. Above a critical interqubit coupling strength, quantum chaos sets in, leading to quantum ergodicity of the computer eigenstates. In this regime the noninteracting qubit structure disappears, the eigenstates become complex and the operability of the computer is destroyed. Despite the fact that the spacing between multi-qubit states drops exponentially with the number of qubits $n$, we show that the quantum chaos border decreases only linearly with $n$. This opens a broad parameter region where the efficient operation of a quantum computer remains possible.

The obtained chaos border for the quantum computer melting induced by inter qubit coupling is of principal importance. Indeed, for $n=1000$, the minimum number of qubits for which Shor's algorithm becomes useful, the multi-qubit spacing becomes $\Delta_n \sim 10^3 \times 2^{-10^3} \Delta_0 \sim 10^{-298}$ K, where we used $\Delta_0 \sim 1$ K that corresponds to the typical one-qubit spacing in the experimental proposals of Vagner et al. and Kane. It is clear that the residual interaction $J$ between qubits in any experimental realization of the quantum computer will be larger than this. For example, in the proposal of Kane, the increase of effective electron mass by a factor of two, induced by the electrostatic gate potential, means that the spin-spin interaction is changed from $J \sim \Delta_0 \sim 1$ K (corresponding to a distance between donors of $200$ Å and an effective Bohr radius of $30$ Å) to the residual interaction $J \sim 10^{-5}$ K $\gg \Delta_n$. However, the quantum chaos border found in [107] is $J_{cs} \approx 0.4 \Delta_0/n \gg \Delta_n$. Only for $J > J_{cs}$ the multi-qubit states start to be mixed while for $\Delta_n \ll J<J_{cs}$ noninteracting multi-qubit states are well defined and the quantum computer can operate successfully.

A pictorial image of the quantum computer melting induced by the coupling between qubits is shown in the color figure on the last back page of the report. Color represents the level of quantum eigenstate entropy $S_q$, with bright red for the maximum values ( $S_q \approx 11$) and blue for the minimal ones ($S_q =0$). For $S_q =0$ the quantum computer eigenstate is represented by one noninteracting multi-qubit state, for $S_q=1$ it is composed from 2 multiqubit states and for $S_q=n$ its is composed from approximately $2^n$ states. Horizontal axis is the energy of the computer eigenstates counted from the ground state to the maximal energy ( $\approx 2n\Delta_0$). Vertical axis is the scaled value of the interqubit coupling $J/\Delta_0$, varying from $0$ to $0.5$. Here $n=12$ and one random realization of interqubit coupling is chosen [107].