Incorporating encoding into quantum system design

When creating a quantum system whose natural dynamics provide useful computational operations, designers have two key tools at their disposal: the (constrained) choice of both the Hamiltonian and the the initial state of the system (an encoding). Typically, we fix the design and utilize encodings post factum to tolerate experimental imperfections. In this paper we describe a vital insight that incorporates encoding into the design process, with radical consequences. This transforms the study of perfect state transfer from the unrealistic scenario of specifying the Hamiltonian of an entire system to the far more realistic situation of being given a Hamiltonian over which we had no choice in the design, and designing time control of just two parameters to still achieve perfect transfer.

Figure 1

A uniformly coupled spin chain (a), described by Hamiltonian B, is extended to fix some of the overall system's eigenvalues (b), permitting a task such as perfect encoded transfer between the two additions. (c) The extensions are simulated by time-varying control of two coupling strengths.

Figure 2

Extending a uniformly coupled chain of 40 qubits to 124 qubits, achieving perfect encoded transfer. We chose certain eigenvalues (circles) to fit the perfect transfer conditions. Even the unfixed ones (triangles) gave a good approximation to those conditions.

Figure 3

(a) Eigenvalues of a chain extended from 40 qubits to give PST. The central eigenvalues satisfy the linear condition (circles), with the others (triangles) being uncontrolled. The continuous (blue) line depicts the weight of each eigenvector on the first site of the chain. (b) Plot of the encoded state transfer fidelity of the extended perfect transfer chain (dashed) compared to original chain (continuous). A plot of the encoded state transfer fidelity using a 124-qubit uniformly coupled chain is indistinguishable from the perfect transfer case at the resolution of this image (Fig. 4 shows how this disparity changes with extension length).

Figure 4

For a central region of 40 qubits, we find symmetric extensions and assess the transfer error at $t=\pi /\delta$ where $\delta$ is fixed. We compare end-to-end transfer (triangles) and optimal encoding over the entire encoding and decoding region (circles). We also include the time-optimized transfer error of a uniformly coupled chain of the same length (diamonds) using encoding over the entire encoding and decoding region.

Figure 5

For a 40-qubit chain as specified in Fig. 2, states can be created on the first 20 qubits of the bulk with a maximum error for a given dimension of space.