Exact quantum decision diagrams with scaling guarantees for Clifford+$T$ circuits and beyond

TL;DR

This work addresses reliable, scalable simulation of Clifford+$T$ quantum circuits using exact algebraic decision diagrams (EVDD and LIMDD) to avoid floating-point errors. It introduces an algebraic representation of edge weights and proves that coefficient sizes are linear in the $T$-count $t$ and number of qubits $n$, while the overall runtime and node counts scale as $2^t \cdot \text{poly}(g, n)$, with $g$ the number of Clifford gates. A key insight is the link between stabilizer nullity and DD width, yielding explicit width bounds: LIMDD width $\le 2^t$ and EVDD width $\le 2^{\min(\#H, 2\cdot\#CZ + \#T)}$, extendable to Toffoli gates. Open-source experiments show exact algebraic methods can outperform floating-point DDs in many cases and provide provable correctness where FP methods fail. Overall, the paper establishes the first scaling guarantees for exact quantum DD simulation with a universal gate set, opening paths to broader applications beyond Clifford+$T$ circuits.

Abstract

A decision diagram (DD) is a graph-like data structure for homomorphic compression of Boolean and pseudo-Boolean functions. Over the past decades, decision diagrams have been successfully applied to verification, linear algebra, stochastic reasoning, and quantum circuit analysis. Floating-point errors have, however, significantly slowed down practical implementations of real- and complex-valued decision diagrams. In the context of quantum computing, attempts to mitigate this numerical instability have thus far lacked theoretical scaling guarantees and have had only limited success in practice. Here, we focus on the analysis of quantum circuits consisting of Clifford gates and $T$ gates (a common universal gate set). We first hand-craft an algebraic representation for complex numbers, which replace the floating point coefficients in a decision diagram. Then, we prove that the sizes of these algebraic representations are linearly bounded in the number of $T$ gates and qubits, and constant in the number of Clifford gates. Furthermore, we prove that both the runtime and the number of nodes of decision diagrams are upper bounded as $2^t \cdot poly(g, n)$, where $t$ ($g$) is the number of $t$ gates (Clifford gates) and $n$ the number of qubits. Our proofs are based on a $T$-count dependent characterization of the density matrix entries of quantum states produced by circuits with Clifford+$T$ gates, and uncover a connection between a quantum state's stabilizer nullity and its decision diagram width. With an open source implementation, we demonstrate that our exact method resolves the inaccuracies occurring in floating-point-based counterparts and can outperform them due to lower node counts. Our contributions are, to the best of our knowledge, the first scaling guarantees on the runtime of (exact) quantum decision diagram simulation for a universal gate set.

Figures (9)

• Figure 1: An example of a quantum circuit with 2 qubits and 6 gates. Simulating this circuit consists of first creating the input state $\ket{\phi_0}=\ket{0}^{\otimes 2}$. Then, iteratively, the states $\ket{\phi_i}$ are updated according to the next gate and replaced by the state $\ket{\phi_{i+1}}$. \ref{['fig:leadingExample']} shows the simulation of this circuit with decision diagrams, where the states $\ket{\phi_i}$ are represented by an EVDD and LIMDD.
• Figure 2: Examples of decision diagram representations for a vector (left). For convenience, the edges pointing to 0 are omitted. Edges in EVDD and LIMDD without label have label 1 resp. $1\cdot I^{\otimes m}$. Figure reproduced from thanos2024automated.
• Figure 3: Motivating example: numerical errors can lead to a larger decision diagram. The figure shows the intermediate EVDDs of the $2$-qubit circuit $H_1 T_1 T_1 S^\dagger_1 H_1 CNOT_{1,2}$ when applied to the input state $\ket{00}$, both with an exact representation of the complex numbers on the edges (above) and with a floating point arithmetic error so that the value $a := \frac{1-\omega^2\cdot(-i)}{1+\omega^2\cdot(-i)}$ is not exactly equal to $0$ (below). Here, $\omega = \frac{1}{\sqrt{2}}(1+i)\approx 0.707+0.707i$.
• Figure 4: Visualization of the proof that each EVDD-edge label is a quotient of state-vector entries. Let $a$ be an bulk edge label in the DD, and let $x_k$ be the parent node of this edge and $x'_{k+1}$ its child node. By the "low"-canonicity rule, every node has at least one outgoing edge with edge label '1'. Therefore, there exist paths from both $x_k$ to the terminal node and from $x'_{k+1}$ to the terminal node with only '1's as edge labels. Concatenating those paths with a path from the root node (i.e., $x_1$) to $x_k$, we have two paths from root to terminal representing the values $\langle x\mid\phi\rangle$ and $\langle y\mid\phi\rangle$ for some $x,y\in\{0,1\}^n$. As these paths differ only by the edge label $a$, we have $a=\frac{\langle x\mid\phi\rangle}{\langle y\mid\phi\rangle}$.
• Figure 5: Example of \ref{['thm:edgelabelbound']}, \ref{['cor:LIMDD-width_mainText']} and \ref{['cor:EVDD-bound']}, showing the intermediate EVDD and LIMDDs of the 2-qubit Clifford$+T$ circuit from \ref{['fig:circuit_example']}. \ref{['thm:edgelabelbound']}: All edge values in the bulk and the magnitude of the root edge value are contained in $R_{2n+2t+1}$ after $t$ T-gates have been applied. (Note that the magnitude of the root edge label for the two rightmost states is $|\frac{1}{4}+\frac{1}{4}\sqrt{2}+\frac{1}{4}i|^2=\frac{1}{4}+\frac{1}{8}\sqrt{2}$, which is in $R_3$.) \ref{['cor:LIMDD-width_mainText']}: The EVDD width is at most $2^{\min({\#H,2\cdot\#CZ + \#T})}$ where $\#G$ is the number of applied $G$-gates for $G\in\{H,CZ,T\}$. \ref{['cor:EVDD-bound']}: The LIMDD width is at most $2^t$ after $t$ T-gates have been applied.

Theorems & Definitions (69)

• corollary 1
• lemma 1: Density matrix entry lemma
• proof : Proof of \ref{['lemma:vector-entry-lemma']}
• definition 1
• theorem 1: DD-representation needs only short numbers.
• proof : Proof of \ref{['thm:edgelabelbound']}
• theorem 2: DD-simulation needs only short numbers.
• proof : Proof sketch
• theorem 3: DD size bounds
• definition 2: stabilizer nullity beverland2020lower