Mimicking time evolution within a quantum ground state: Ground-state quantum computation, cloning, and teleportation

Ground-state quantum computers mimic quantum-mechanical time evolution within the amplitudes of a time-independent quantum state. We explore the principles that constrain this mimicking. A no-cloning argument is found to impose strong restrictions. It is shown, however, that there is flexibility that can be exploited using quantum teleportation methods to improve ground-state quantum computer design.

Figure 1

Quantum dot realization of ground-state quantum computer with a single qubit. An example is shown for $N=4$, where the electronic wave function (2) purely for illustration is taken to be ${\left[10\sqrt{1∕2}\sqrt{1∕2}010110\right]}^{†}∕\sqrt{5}$ and the shading indicates a nonzero expectation value of the electronic charge density.

Figure 2

Comparison of (a) classical digital circuit, (b) $\mathrm{GSQC}$ implemented in an array of quantum dots, and (c) conventional time-dependent quantum computer realized with charge-based quantum dot qubits. In (a), logical value 1 corresponds to $10\mathrm{V}$. In (b), the expectation value of the ground-state charge density is indicated by shading of the quantum dots; it is analogous to the pattern of voltages in (a). In (c), an electron shifts in time coherently between two dots in each qubit. Only the final state is depicted.

Figure 3

(a) A many-body realization of a single qubit. Five [$(N+1)=5$ in this example] electrons are each confined to a row of three dots. The total Hilbert space has dimension $3^{N+1}=3^5$, but only $2(N+1)=10$ states are computationally meaningful. One such state, $\mid 0_3⟩$, is portrayed here. The state has four electrons in their “idle” states and electron $i=3$ (the fourth electron from the top) in the logical 0 state. (b) The same state in the single electron implementation of Fig. 1.

Figure 4

A controlled-NOT gate in many-body-qubit realization of a $\mathrm{GSQC}$. Each row of three quantum dots in qubit $A$ contains a single electron, as does each row of three quantum dots in qubit $B$. Within each dotted line, however, there are five dots instead of three that contain a single electron. The computer executes a controlled-NOT operation between row $j=1$ and row $j=2$ using the Hamiltonian (7), just as in the previous implementation.

Figure 5

The spectrum of the Hamiltonian (14) is that of two chains with on-site potentials $2ϵv_i$ and tunneling matrix elements $ϵt_i$. At the ends of each chain, the on-site potential is $ϵv_i$ rather than $2ϵv_i$, and a small perturbation $\pm ϵv_0\Delta$ breaks the equivalence of the chains, so that the system has a single nondegenerate eigenstate. The constant energy factor $ϵ$ is omitted from the figure.

Figure 6

Three choices of $v_i$ and the corresponding values of ${\lambda }_i$. In each case, the filled circles give values of $v_i$ and the empty circles give values of ${\lambda }_i$. The dashed lines are guides to the eye of the form of the ${\lambda }_i$. (a) A minimum in the $v_i$ leads to an energy gap in accordance with scalability criterion, (ii) but the ${\lambda }_i$ are substantial only in a localized region of $i$. (b) If the minimum is near $i=0$, then $\mid {\lambda }_0\mid$ large but $\mid {\lambda }_N\mid$ is small, conflicting with (iii). (c) If the minimum is near $i=N$, then $\mid {\lambda }_N\mid$ large but $\mid {\lambda }_0\mid$ is small, conflicting with (i).

Figure 7

Ground-state computer that applies gates by quantum teleportation to produce $U_N\cdots U_1\mid 0⟩$. An example is shown for $N=3$ operations, where the protocol entails five qubits. The seven rows of the computer execute steps (a)–(g) in sequence. Rows 1–3 yield two EPR pairs, assuming that all qubits are input with logical 0. Row 4 applies the unitary gates $U_i$. Rows $5–7$ measure four of the five qubits, and the desired state resides with some probability in the final row of qubit $E$. To increase $N$, more qubits are added and the same pattern of gates is employed.