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arXiv:2603.20376

Parameter-optimal unitary synthesis with flag decompositions

Korbinian Kottmann, David Wierichs, Guillermo Alonso-Linaje, Nathan Killoran

quant-phSubmitted Mar 20, 2026• Announced Mar 24, 2026• 0 citations
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Abstract

We introduce the flag decomposition as a central tool for unitary synthesis. It lets us carve out a diagonal unitary with $2^n$ degrees of freedom in such a way that the remaining flag circuit is parametrized by the optimal number of $4^n-2^n$ rotations. This enables us to produce parameter-optimal quantum circuits for generic unitaries and matrix product state preparation. Our approach improves upon the state of the art, both when compiling down to the {Clifford + Rot} gate set with what we call selective de-multiplexing, and when using phase gradient resource states together with quantum arithmetic to implement multiplexed rotations. All of our synthesis methods are efficiently implementable in terms of recursive Cartan decompositions realized by standard linear algebra routines, making them applicable to all practically relevant system sizes.

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We introduce the flag decomposition as a central tool for unitary synthesis. It lets us carve out a diagonal unitary with 2n2^n2n degrees of freedom in such a way that the remaining flag circuit is parametrized by the optimal number of 4n−2n4^n-2^n4n−2n rotations. This enables us to produce parameter-optimal quantum circuits for generic unitaries and matrix product state preparation. Our approach improves upon the state of the art, both when compiling down to the {Clifford + Rot} gate set with what we call selective de-multiplexing, and when using phase gradient resource states together with quantum arithmetic to implement multiplexed rotations. All of our synthesis methods are efficiently implementable in terms of recursive Cartan decompositions realized by standard linear algebra routines, making them applicable to all practically relevant system sizes.
We show the applicability of the Cartan decomposition of Lie algebras to quantum circuits. This approach can be used to synthesize circuits that can efficiently implement any desired unitary operation. Our method finds explicit quantum circuit representations of the algebraic generators of the relevant Lie algebras allowing the direct implementation of a Cartan decomposition on a quantum computer. The construction is recursive and allows us to expand any circuit down to generators and rotation matrices on individual qubits, where through our recursive algorithm we find that the generators themselves can be expressed with controlled-not (cnot) and swap gates explicitly. Our approach is independent of the standard cnot implementation and can be easily adapted to other cross-qubit circuit elements. In addition to its versatility, we also achieve near-optimal counts when working with cnot gates, achieving an asymptotic cnot cost of 21164n\frac{21}{16}{4}^{n}1621​4n for nnn qubits.
Resource-efficient and high-precision approximate synthesis of quantum circuits expressed in the Clifford+T gate set is vital for Fault-Tolerant quantum computing.Efficient optimal methods are known for single-qubit R Z unitaries, otherwise the problem is generally intractable.Search-based methods, like simulated annealing, empirically generate low resource cost approximate implementations of general multi-qubit unitaries so long as low precision (Hilbert-Schmidt distances of ε ≥ 10 -2 ) can be tolerated.These algorithms build up circuits that directly invert target unitaries.We instead leverage search-based methods to first approximately diagonalize a unitary, then perform the inversion analytically.This lets difficult continuous rotations be bypassed and handled in a postprocessing step.Our approach improves both the implementation precision and run time of synthesis algorithms by orders of magnitude when evaluated on unitaries from real quantum algorithms.On benchmarks previously synthesizable only with analytical techniques like the Quantum Shannon Decomposition, diagonalization uses an average of 95% fewer non-Clifford gates.
Near-term quantum computers have significant error rates and short coherence times, so compilation of circuits to be as short as possible is essential. Two types of compilation problems are typically considered: circuits to prepare a given state from a fixed input state, called “state preparation”; and circuits to implement a given unitary operation, for example by “unitary synthesis”. In this paper we solve a more general problem: the transformation of a set of m states to another set ofmmmstates, which we call “multi-state preparation”. State preparation and unitary synthesis are special cases; for state preparation, m=1, while for unitary synthesis, m is the dimension of the full Hilbert space. We generate and optimize circuits for multi-state preparation numerically. In cases where a top-down approach based on matrix decompositions is also possible, our method finds circuits with substantially (up to 40 %) fewer two-qubit gates. We discuss possible applications, including efficient preparation of macroscopic superposition (“cat”) states and synthesis of quantum channels.
We present a simple algorithm that implements an arbitrary n -qubit unitary operator using a Clifford + T circuit with T-count O ( 2 4 n / 3 n 2 / 3 ) and ancilla-count O ( 2 2 n / 3 n 1 / 3 ) . The previous best T-count for unitary synthesis was O ( 2 3 n / 2 n ) with O ( 2 n / 2 ) ancillae by [Low , ]. There is a fundamental tradeoff between T gates and ancillae, and our algorithm extends the achievable regime of this tradeoff, reducing the exponent in T-count from 3 n / 2 to 4 n / 3 . The best known lower bound remains Ω ( 2 n ) , so the optimal T-count for general unitary synthesis is still open. Our construction is based on a recursive application of the cosine-sine decomposition, together with a generalization of the optimal diagonal unitary synthesis method by [D. Gosset , ] to multi-controlled k -qubit unitaries.
This tutorial builds upon the foundations established in Part~II to present a unified, implementation-oriented overview of quantum state initialization and unitary decomposition for n -qubit systems, available in QFlux . We begin with the preparation of arbitrary quantum states and develop two complementary constructions: (i) a recursive multiplexor method (after Shende et al.) that disentangles qubits via multiplexed R y and R z rotations, and (ii) an algebraic scheme based on uniformly controlled rotations (UCRs) (after Möttönen et al.) that realizes the same mapping through analytically defined rotation networks with predictable gate counts. We then extend these state-preparation tools to generic unitary synthesis through Givens-rotation, column-by-column, and recursive cosine-sine (CSD) and quantum Shannon (QSD) decompositions, explicitly linking linear-algebraic factorizations to executable circuits while tracking CNOT complexity. Finally, we introduce the Walsh decomposition for diagonal unitaries and show how Gray-code ordering and local CNOT cancellations yield an O(2^n) entangling-gate cost, providing shallow, NISQ-friendly implementations. Taken together, these techniques form a pedagogical bridge from matrix analysis to hardware-efficient circuit constructions, offering clear design rules, closed-form parameters, and scalable synthesis pathways for simulation and experiment.
As a model for quantum computation, quantum circuits have found their wide applications in communications, cryptography and information processing. In order to synthesize arbitrary quantum circuits, we present a new type of gate called quantum multiplex rotation gate, which is implemented by simply elementary gates. A method based on QR decomposition and two optimization rules are proposed to decompose general quantum circuit acting on n-qubits into quantum multiplex rotation gates. In comparison with other synthesis algorithms by QR decomposition, our methods achieve better performance in terms of elementary gate counts, 1.2 × 4n approximately.