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

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.

💡 Research Summary

The paper addresses a long‑standing problem in quantum circuit simulation: decision diagrams (DDs) such as EVDD and LIMDD suffer from floating‑point inaccuracies that cause node duplication or erroneous merges, dramatically inflating memory and runtime. The authors propose to replace floating‑point edge weights with an exact algebraic representation of complex numbers tailored to the Clifford+T gate set. They prove that every amplitude (or, equivalently, every entry of the density matrix multiplied by 2ⁿ) produced by an n‑qubit circuit containing t T‑gates can be expressed as a linear combination of a small fixed basis {1, i, √2, (1±i)/√2} with integer coefficients whose bit‑length grows only linearly with t and n, and is independent of the number of Clifford gates g. Consequently, the algebraic objects stored on DD edges have size O(t + n).

The authors then show how these algebraic expressions can be directly attached to the edges of EVDD and LIMDD. Because the representation is exact, the canonical merging criteria of the DDs (equality up to a scalar for EVDD, equality up to a scalar and local Pauli operators for LIMDD) are satisfied without any numerical tolerance. This eliminates the source of error accumulation entirely.

A central technical insight is the connection between the width of a DD and the stabilizer nullity of the quantum state it represents. Stabilizer nullity is invariant under Clifford operations and increases by at most one for each T‑gate. Since the DD width is bounded by a function of the nullity, the total number of nodes after simulating a Clifford+T circuit is bounded by O(2ᵗ·poly(g,n)). In particular, LIMDDs for the final state require at most O(2ᵗ) nodes, while EVDDs admit a slightly weaker bound that depends on the minimum of the number of H‑gates and the combined count of CZ and T gates. The authors also extend the analysis to the universal Clifford+Toffoli set, showing that the same asymptotic behavior holds.

To validate the theory, the paper presents an open‑source extension of the Q‑Sylvan DD library that implements the algebraic edge representation. Benchmarks on standard quantum circuits (including error‑correction codes and small algorithms) demonstrate that the exact method not only avoids the numerical blow‑up seen in floating‑point EVDD implementations but also often uses fewer nodes and runs faster. In cases where the floating‑point approach fails due to memory exhaustion or loss of fidelity, the exact method succeeds.

The contributions can be summarised as follows: (1) an exact algebraic encoding of all complex coefficients arising from Clifford+T circuits, with size linear in the T‑count and qubit number; (2) rigorous proofs that DD node count and runtime are bounded by 2ᵗ·poly(g,n), establishing fixed‑parameter tractability with respect to the T‑count; (3) a practical implementation that outperforms existing floating‑point DD simulators and demonstrates the feasibility of exact DD‑based quantum simulation. This work therefore provides the first theoretical scaling guarantees for exact DD simulation of a universal gate set and opens the door to reliable, high‑performance DD techniques for a broader class of continuous‑valued applications.