A Knowledge Compilation Map for Quantum Information

TL;DR

The paper develops a knowledge compilation map for quantum state representations by analytically comparing matrix product states (MPS), restricted Boltzmann machines (RBM), and quantum decision diagrams (QDDs) including LIMDD and SLDD_x variants. It introduces rapidity to compare non-canonical data structures and proves that MPS is at least as rapid as SLDD_x across key operations, while MPS is strictly more succinct than SLDD_x and ADD, and LIMDD is exponentially more succinct than SLDD_x yet still less rapid than MPS. The authors also establish a series of concrete separations using the Sum state to show MPS and LIMDD outperform SLDD_x in succinctness, with RBM incomparably related to the QDD families and subject to representational trade-offs in practice. They prove hardness results for fidelity in LIMDD and RBM under ETH, and provide transformations between QDDs and to MPS, enabling a unified view of time-space efficiency for quantum circuit simulation, variational quantum algorithms, and verification tasks.

Abstract

Quantum computing is finding promising applications in optimization, machine learning and physics, leading to the development of various models for representing quantum information. Because these representations are often studied in different contexts (many-body physics, machine learning, formal verification, simulation), little is known about fundamental trade-offs between their succinctness and the runtime of operations to update them. We therefore analytically investigate three widely-used quantum state representations: matrix product states (MPS), decision diagrams (DDs), and restricted Boltzmann machines (RBMs). We map the relative succinctness of these data structures and provide the complexity for relevant query and manipulation operations. Further, to chart the balance between succinctness and operation efficiency, we extend the concept of rapidity with support for the non-canonical data structures studied in this work, showing in particular that MPS is at least as rapid as some DDs. By providing a knowledge compilation map for quantum state representations, this paper contributes to the understanding of the inherent time and space efficiency trade-offs in this area.

Figures (9)

• Figure 1: The $3$-qubit GHZ state $1/\sqrt 2(\ket{000} + \ket{111})$, displayed using different data structures. The unlabelled edges for ADD, $\textsf{SLDD}_{\times}$, LIMDD have resp. label 1, 1, $\mathbb{I}_{}\xspace$. In the RBM, the weights of edges incident to $h_1,h_2$ ($h_3, h_4$) are all $i\pi/3$ ($-i\pi/3$); the hidden node biases $(\beta_{h_1}, \beta_{h_2}, \beta_{h_3}, \beta_{h_4}) = i\pi \cdot (1/3, 2/3, -1/3, -2/3)$; the visible node biases $\alpha_{v_1}=\alpha_{v_2}=\alpha_{v_3}=0$.
• Figure 2: Succinctness relations between various classical data structures for representing quantum states. Solid arrows $A\to B$ denote $B\prec_s\xspace A$, i.e., $B$ is strictly more succinct than $A$. Crossed arrows $A\space\mathclap{\longrightarrow}{\space\times}\space B$ denote a separation $B \npreceq_{s}\xspace A$; a bidirectional crossed arrow implies incomparability. Blue arrows indicate novel relations that we identified.
• Figure 4: Visualization of the proof of \ref{['thm:sufficient-condition-rapidity']} in case $OP$ is a transformation operation: Given runtime monotonic algorithm $ALG_2^{rm}$ implementing $OP$ on language $D_2$, the composed algorithm $ALG_1 \,\triangleq\, g \circ ALG_2^{rm} \circ f$ for $OP(D_1)$ is at least as rapid as $ALG_2$. To prove this, we consider $x_2\in D_2$ and an equivalent and at most only polynomially larger than $x_1\in D_1$ and show that $ALG_1$ takes at most polynomially more time on $x_1$ than $ALG_2$ on $x_2$. $ALG_1$ is also runtime monotonic. Horizontally-aligned instances of data structures are equivalent, i.e. represent the same quantum state.
• Figure 6: Rapidity relations between data structures considered here. A solid arrow $D_1\to D_2$ means $D_2$ is at least as rapid as $D_1$ for all operations satisfying \ref{['i:omega']} and \ref{['i:rm']} of \ref{['thm:sufficient-condition-rapidity']}.
• Figure 7: $3$-qubit quantum circuit preparing the GHZ state.

Theorems & Definitions (111)

• Definition 1: Inspired by fargier2014knowledge
• Definition 2: Isomorphic nodes
• Theorem 1
• Theorem 2
• Theorem 3
• Definition 3: Rapidity for non-canonical data structures
• Theorem 4
• Definition 4: Weakly minimizing transformation
• Definition 5: Runtime monotonic algorithm
• Theorem 5: A sufficient condition for rapidity