Efficient algorithm for multiqudit twirling for ensemble quantum computation

We present an efficient algorithm for twirling a multiqudit quantum state. The algorithm can be used for approximating the twirling operation in an ensemble of physical systems in which the systems cannot be individually accessed. It can also be used for computing the twirled density matrix on a classical computer. The method is based on a simple nonunitary operation involving a random unitary. When applying this basic building block iteratively, the mean squared error of the approximation decays exponentially. In contrast, when averaging over random unitary matrices the error decreases only algebraically. We present evidence that the unitaries in our algorithm can come from a very imperfect random source or can even be chosen deterministically from a set of cyclically alternating matrices. Based on these ideas we present a quantum circuit realizing twirling efficiently.

Figure 1

Twirling can be realized by the repeated application of this basic building block where $U_k$ is a random unitary generated for the $k$th iteration or a unitary chosen from a cyclically alternating set of unitaries. $M$ represents measurement in the computational basis.

Figure 2

(Color online) Mean squared error for the recursive method with random matrices when applied on three-qubit states. We plot (dotted) the average over $10000$ realizations and (solid line) the theoretical prediction computed from Eq. (27). For better visibility, the error is shown only for every second iteration.

Figure 3

(Color online) Time dependence of the error for (a) two and (b) three qubits for the deterministic method using two and three unitaries, respectively. For better visibility, the error is shown only for every second iteration. Dashed line indicates the error for the method using random matrices, given in Eq. (27).

Figure 4

(Color online) Time dependence of the normalized error of the density matrix for bipartite twirling for $d$=(a) 8 and (b) 16. In both cases 25 realizations are shown. See text for the algorithm used for generating $d$-dimensional unitaries. For better visibility, the error is shown only for every second iteration. Dashed line indicates the error for the method using random matrices uniformly distributed over $\mathrm{U}\left(d\right)$.