Constructing general unitary maps from state preparations

We present an efficient algorithm for generating unitary maps on a $d$-dimensional Hilbert space from a time-dependent Hamiltonian through a combination of stochastic searches and geometric construction. The protocol is based on the eigendecomposition of the map. A unitary matrix can be implemented by sequentially mapping each eigenvector to a fiducial state, imprinting the eigenphase on that state, and mapping it back to the eigenvector. This requires the design of only $d$ state-to-state maps generated by control wave forms that are efficiently found by a gradient search with computational resources that scale polynomially in $d$. In contrast, the complexity of a stochastic search for a single wave form that simultaneously acts as desired on all eigenvectors scales exponentially in $d$. We extend this construction to design maps on an $n$-dimensional subspace of the Hilbert space using only $n$ stochastic searches. Additionally, we show how these techniques can be used to control atomic spins in the ground-electronic hyperfine manifold of alkali metal atoms in order to implement general qudit logic gates as well to perform a simple form of error correction on an embedded qubit.

Figure 1

(Color online) The hyperfine structure of ${}^{133}\text{C}\text{s}$ in the $6S_{1/2}$ ground state. Microwaves and rf magnetic fields provide controllable dynamics on the 16-dimensional Hilbert space. A detuned laser light shift can be used to create a relative phase between the $F=4$ and $F=3$ manifolds. By considering controls on the highlighted subspace of states, we recover a system that satisfies the criteria proposed in Sec. 2.

Figure 2

(Color online) Optimized control fields for implementing the seven-dimensional Fourier transform on the $F=3$ hyperfine manifold in ${}^{133}\text{C}\text{s}$. The duration of the pulse is less than 1.2 ms and yields a unitary map that has an overlap of 0.9854 with the desired target. As an example, we show the action of the resulting unitary on the $Z$ eigenstates of angular momentum. The conjugate variable of $F_z$ is the azimuthal angle $\varphi$. If we Fourier transform a $Z$ eigenstate, a state with a well-defined value of $F_z$, we obtain a state that has a well-defined value of $\varphi$: a squeezed state.

Figure 3

(Color online) (a) A schematic of the error correction protocol we have designed using subspace maps. We track the basis elements of our encoded subspace; here $|0⟩$ is red (aligned right) and $|1⟩$ is blue (aligned left), via their populations in the $x$ and $z$ bases. (i) The initial embedded qubit we wish to protect is in a superposition of the $|4,4_z⟩$ and $|3,3_z⟩$ states. (ii) We use subspace maps to encode the state in the basis $|3,3_x⟩$, $|3,-3_x⟩$. (iii) In this basis, a small $z$ rotation shifts population into the states $|3,2_x⟩$ and $|3,-2_x⟩$. (iv) Using subspace maps, we can transfer the small population that has left the encoded space in the two states $|4,4_z⟩$ and $|4,-4_z⟩$. Now we can perform a nondemolition measurement of the total angular momentum $F$. If $F=3$, we can be certain our state lies in the encoded subspace (ii). If we measure $F=4$, the system is in configuration (v), which we then conditionally transform back to the encoded state. In (b), we examine the performance of the error correction. On the $x$ axis, we have the angle of rotation in the $z$ direction due to the magnetic field error. On the $y$ axis is the fidelity between the initial and post-error states, as average over pure states drawn from the Haar measure. The blue line (broad plateau) shows the fidelity of the error-corrected states and the green (oscillating function) shows the fidelity if the state had simply stayed in the subspace $|4,4_z⟩$, $|3,3_z⟩$.