Parallelism for quantum computation with qudits

Robust quantum computation with $d$-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high-fidelity two-qudit gates. We first describe how to implement parallel single-qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth (time) of the parallel gate sequence is equal to the height of an associated tree. The implementation of a given unitary can then optimize the tradeoff between gate time and resources used. These ideas are illustrated for qudits encoded in the ground hyperfine states of the alkali-metal atoms ${}^{87}\mathrm{Rb}$ and ${}^{133}\mathrm{Cs}$. Second, we provide a protocol for implementing parallelized nonlocal two-qudit gates using the assistance of entangled qubit pairs. Using known protocols for qubit entanglement purification, this offers the possibility of high-fidelity two-qudit gates.

Figure 1

A single $d=8$ qudit encoded in the ground-state hyperfine levels of ${}^{87}\mathrm{Rb}$. A pair of lasers can couple states in different hyperfine manifolds according to the selection rule $\mathrm{\Delta }{M}_F=0,\pm 1$.

Figure 2

Coupling graph for ${}^{87}\mathrm{Rb}$. There is an edge between two states if a pair of lasers can couple them as in Fig. 1.

Figure 3

A spanning tree (left) and a BCT (right) for node 3 of ${}^{87}\mathrm{Rb}$. On the left, the graph in Fig. 2 is redrawn with state 3 at the top, omitting the edges (0,6) and (5,1) in order to remove cycles and make the graph a tree with two leaves, corresponding to $\mid 7⟩$ and $\mid 6⟩$. We obtain the BCT on the right by replacing each nonleaf by a leaf connected to a chain of $p$ nodes, where $p$ is the number of children. For example, node 2 in the spanning tree has two children, so at right it is replaced with nodes $2$, $2a$, and $2b$. We then attach one child to each of the new nodes. The result is that, working from the bottom up, rotations on the same level [for example, (2,5) and (4,1)] can be done simultaneously, accomplishing synthesis of $\mid 3⟩$.

Figure 4

(Color online) State synthesis timing diagram for $\mid 3⟩$ for the rubidium alkali-metal atom using two-way parallelism. The time steps, indicated in bold, are taken from the BCT in Fig. 3, working from the bottom up. All transitions are directed toward $\mid 3⟩$.

Figure 5

(Color online) Coupling graph for ${}^{133}\mathrm{Cs}$.

Figure 6

(Color online) State-synthesis timing diagram for $\mid 1⟩$ for the cesium alkali-metal atom using two-way parallelism. All transitions are directed toward $\mid 1⟩$.

Figure 7

(Color online) State-synthesis timing diagram for $\mid 7⟩$ for the cesium alkali-metal atom using two-way parallelism.

Figure 8

A nonlocal two-qudit gate $U=W^{†}$ that realizes the state synthesis $U\mid 0{⟩}_{A,B}=\mid \psi {⟩}_{A,B}$ on qudits $A$ and $B$ using $d-1$ ancillary qubit pairs (indicated by sawtooth lines) each prepared in the state $\mid {\varphi }^{+}{⟩}_{A_j,B_j}=1∕\sqrt{2}(\mid 00⟩+\mid 11⟩{)}_{A_j,B_j}$. Each qubit $A_j\left(B_j\right)$ in the entangled resource can constitute a new particle or a distinct degree of freedom of qudit $A\left(B\right)$. Controlled-NOT gates between $A$ and $A_j$ are conditioned on the basis state $\mid j{⟩}_A$, as indicated by the shading of the control bubble. Double lines denote classical controlled operations dependent on qubit measurement outcomes, $H=e^{i\pi ({\sigma }^x+{\sigma }^z)∕2\sqrt{2}}$, and $P_j=e^{i\pi \mid j⟩⟨j\mid }$. The sequence of steps that can be implemented in parallel is indicated at the bottom.