Classifying quantum data by dissipation

We investigate a general class of dissipative quantum circuit capable of computing arbitrary conjunctive normal form (CNF) Boolean formulas. In particular, the clauses in a CNF formula define a local generator of Markovian quantum dynamics which acts on a network of qubits. Fixed points of this dynamical system encode the evaluation of the CNF formula. The structure of the corresponding quantum map partitions the Hilbert space into sectors, according to decoherence-free subspaces (DFSs) associated with the dissipative dynamics. These sectors then provide a natural and consistent way to classify quantum data (i.e., quantum states). Indeed, the attractive fixed points of the network allow one to learn the sector(s) for which some particular quantum state is associated. We show how this structure can be used to dissipatively prepare quantum states (e.g., entangled states) and outline how it may be used to generalize certain classical computational learning tasks.

Figure 1

Three input qubits $q_{i_j}$ (yellow, top), coupled dissipatively to an ancilla qubit $a$ (blue, bottom), via a four-local Lindbladian (represented by the four solid lines and the solid dot in the center).

Figure 2

Illustration of a dissipative computational network, capable of computing a 3-CNF with $N$ clauses on $n$ variables. The $q_i$ represent input qubits, and the $a_i$ are ancilla qubits (where the $i\mathrm{th}$ clause is evaluated). In this particular example, we show explicitly the clauses $C_1=l_1\vee l_2\vee l_k$ and $C_N=l_k\vee l_j\vee l_n$, where $l_i$ is either $x_i$ or $\neg x_i$.

Figure 3

Example of a network learning the conjunction $f=x_1\wedge \neg x_3$ for $n=5$. Ancilla qubits are in blue, labeled $a_i$, and the $n$ data qubits are labeled by $q_i$. The initial network (top) is configured to compute $x_1\wedge \neg x_1\wedge \cdots \wedge x_5\wedge \neg x_5$ as described by the learning algorithm in the main text. The top row of ancilla qubits (odd) computes the literals $x_i$, and the bottom row (even) computes the negations $\neg x_i$. After training for a sufficiently long time that the network has converged, the final network is given by the bottom diagram, which now has just two ancilla qubits (blue), corresponding to the computation of $x_1\wedge \neg x_3$. The Hilbert space therefore has four associated DFSs. Input (data) qubits are shown in yellow.

Figure 4

Distance to maximally entangled Bell state $|{\psi }^{+}\rangle =\frac{1}{\sqrt{2}}\left(\right|00\rangle +|11\rangle )$ under dissipative evolution for time $t$ (in units of $1/\gamma$). Here, ${\rho }_{1,1}$ is the state after postselecting on measuring 1 in both ancilla qubits [Eq. (39)], which occurs with probability $1/2$. The initial state is the product state $|+,+\rangle$. We use three different norms to evaluate the distance [35]. Inset: the connectivity of the 2+2 qubit dissipative network. $q_{1,2}$ label the two qubits which we wish to entangle, and $a_{1,2}$ are the ancillae used in our scheme. The solid dots represent the Lindbladian connectivity, labeled respectively as ${\mathcal{L}}_{1,2}$.