Learning shallow quantum circuits

TL;DR

This work presents a polynomial-time classical algorithm for learning the description of any unknown n-qubit shallow quantum circuit U (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of U.

Abstract

Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown $n$-qubit shallow quantum circuit $U$ (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of $U$. We also provide a polynomial-time classical algorithm for learning the description of any unknown $n$-qubit state $\lvert ψ\rangle = U \lvert 0^n \rangle$ prepared by a shallow quantum circuit $U$ (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of $\lvert ψ\rangle$. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate.

Figures (7)

• Figure 1: Learning geometrically-local shallow quantum circuits. (a) In this example, the geometry is a 2D lattice where each vertex has a degree at most 4. The lightcone of the blue qubit (for depth $d=2$) is the union of the blue and orange qubits. (b) The learned circuit acts on an extended geometry with $2n$ qubits, where each system qubit (black) is attached to an ancilla qubit (red). Note that each ancilla qubit is connected only with its corresponding system qubit (red edges).
• Figure 2: A coloring of $k$-dimensional lattice with $k+1$ colors, where different regions of the same color are separated by distance at least $R$. (a) A coloring of 2-dimensional lattice. (b) A coloring of 3-dimensional lattice (the fourth color is not shown).
• Figure 3: Efficient learning of quantum states generated by a shallow circuit in 1D. For each local region $A,B,C,\dots$ we find a list of local inversion circuits, and merge them together by solving a constraint satisfaction problem.
• Figure 4: Learning to disentangle a quantum state generated by a shallow circuit in 2D. (a) The middle region $M$ can be inverted by solving a similar 1D constraint satisfaction problem as in Fig. \ref{['fig:1dlearning']}. (b) After inverting all the gray $B_i$ regions, the remaining white $A_i$ regions are disentangled into a tensor product of pure states.
• Figure 5: Each of the states on the white $A_i$ regions in Fig. \ref{['fig:2dlearning']} (b) can be viewed as being prepared by a depth-$2d$ circuit acting on $A_i$ (white) as well as ancilla qubits $A_i^L$ and $A_i^R$ (blue).

Theorems & Definitions (115)

• Theorem : Summary of main results
• Theorem 1: Learning shallow quantum circuits; see Theorem \ref{['thm:shallow-SU4-gates']}
• Theorem 2: Learning geometrically-local shallow circuits; see Theorem \ref{['thm:geo-kD-lattice-optimized']}
• Theorem 3: Learning shallow circuits with quantum queries; see Theorem \ref{['thm:geo-finite-gates']}
• Theorem 4: Learning quantum states prepared by 2D shallow circuits; see Theorem \ref{['thm:2dstate']}
• Definition 1: Reduced channel
• Definition 2: Fidelity
• Definition 3: Average-case distance
• Proposition 1: Normalized Frobenius norm
• proof