Unitary Synthesis, the decomposition of a unitary matrix into a sequence of quantum gates, is a fundamental challenge in quantum compilation. Prevailing reinforcement learning (RL) approaches are often hampered by sparse reward signals, which necessitate complex reward shaping or long training times, and typically converge to a single policy, lacking solution diversity. In this work, we propose QFlowNet, a novel framework that learns efficiently from sparse signals by pairing a Generative Flow Network (GFlowNet) with Transformers. Our approach addresses two key challenges. First, the GFlowNet framework is fundamentally designed to learn a diverse policy that samples solutions proportional to their reward, overcoming the single-solution limitation of RL while offering faster inference than other generative models like diffusion. Second, the Transformers act as a powerful encoder, capturing the non-local structure of unitary matrices and compressing a high-dimensional state into a dense latent representation for the policy network. Our agent achieves an overall success rate of 99.7% on a 3-qubit benchmark(lengths 1-12) and discovers a diverse set of compact circuits, establishing QFlowNet as an efficient and diverse paradigm for unitary synthesis.
One of the challenges in quantum computing is the synthesis of unitary operators into quantum circuits with polylogarithmic gate complexity. Exact synthesis of generic unitaries requires an exponential number of gates in general. We propose a novel approximate quantum circuit synthesis technique by relaxing the unitary constraints and interchanging them for ancilla qubits via block-encodings. This approach combines smaller block-encodings, which are easier to synthesize, into quantum circuits for larger operators. Due to the use of block-encodings, our technique is not limited to unitary opera…
Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, we show that $2(N-1)$ generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the $y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types which are defined in this paper are sufficient to simulate a controlled-unitary gate $\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling on $\mathbb{C}^2\otimes \mathbb{C}^N$. In the scenario of the unitary gate $\mathcal{U}_{cd(M\otimes N)}$ with $M\geq3$ that is locally equivalent to a diagonal uni…
We perform optimal-control-theory calculations to determine the minimum number of two-qubit CNOT gates needed to perform quantum state preparation and unitary operator synthesis for few-qubit systems. By considering all possible gate configurations, we determine the maximum achievable fidelity as a function of quantum circuit size. This information allows us to identify the minimum circuit size needed for a specific target operation and enumerate the different gate configurations that allow a perfect implementation of the operation. We find that there are a large number of configurations that …
We developed a general framework for synthesizing target gates by using a finite set of basic gates, which is a crucial step in quantum compilation. When approximating a gate in SU($n$), a naive brute-force search requires a computational complexity of $O(1/\varepsilon^{(n^2 - 1)})$ to achieve an approximation with error $\varepsilon$. In contrast, by using our method, the complexity can be reduced to $O(-n^2 \log\varepsilon/\varepsilon^{((n^2 - 1)/2)})$. This method requires almost no assumptions and can be applied to a variety of gate sets, including Clifford+$T$ and Clifford+$V$. Further, w…
Whereas quantum complexity theory has traditionally been concerned with problems arising from classical complexity theory (such as computing boolean functions), it also makes sense to study the complexity of inherently quantum operations such as constructing quantum states or performing unitary transformations. With this motivation, we define models of interactive proofs for synthesizing quantum states and unitaries, where a polynomial-time quantum verifier interacts with an untrusted quantum prover, and a verifier who accepts also outputs an approximation of the target state (for the state sy…
The Quantum State Preparation problem aims to prepare an $n$-qubit quantum state $|ψ_v\rangle =\sum_{k=0}^{2^n-1}v_k|k\rangle$ from the initial state $|0\rangle^{\otimes n}$, for a given unit vector $v=(v_0,v_1,v_2,\ldots,v_{2^n-1})^T\in \mathbb{C}^{2^n}$ with $\|v\|_2 = 1$. The problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth and size complexity remain open when ancillary qubits are available. In this paper, we study efficient constructions of quantum circuits with $m$ ancillary qubits that can prepare…
Asymptotically Optimal Circuit Depth for Diagonal Unitary Synthesis and Compilation on Two-Dimensional Grids
School of Computer Science and Engineering, Southeast University（东南大学计算机科学与工程学院）；College of Information Engineering, China Jiliang University（中国计量大学信息工程学院）；Key Laboratory of Computer Network and Information Integration (Southeast University), Ministry of Education（教育部计算机网络与信息集成重点实验室（东南大学））；Nanjing Meteorology Bereau（南京市气象局）
Recently, the dynamics simulation of light-harvesting complexes as an open quantum system, in the weak and strong coupling regimes, has received much attention. In this paper, we investigate a digital quantum simulation approach of the Fenna-Matthews-Olson (FMO) photosynthetic pigment-protein complex surrounded with a Markovian bath, i.e., memoryless, based on a nuclear magnetic resonance (NMR) quantum computer. For this purpose, we apply the decoupling(recoupling) method, which is turn off(on) the couplings and also Solovay-Kitaev techniques to decompose Hamiltonian and Lindbladians into effi…
Communication Complexity of Distributed Unitary Synthesis
State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；School of Computer Science and Technology, University of Chinese Academy of Sciences（中国科学院大学计算机科学与技术学院）
An approach is presented for robustness analysis and quantum (unitary) control synthesis based on the classic method of averaging. The result is a multicriterion optimization competing the nominal (uncertainty-free) fidelity with a well known robustness measure: the size of an interaction (error) Hamiltonian, essentially the first term in the Magnus expansion of an interaction unitary. Combining this with the fact that the topology of the control landscape at high fidelity is determined by the null space of the nominal fidelity Hessian, we arrive at a new two-stage algorithm. Once the nominal …
We present a method that outputs a sequence of simple unitary operations to prepare a given quantum state that is a generalized coherent state. Our method takes as inputs the expectation values of some relevant observables on the state to be prepared. Such expectation values can be estimated by performing projective measurements on $O(M^3 \log(M/δ)/ε^2)$ copies of the state, where $M$ is the dimension of an associated Lie algebra, $ε$ is a precision parameter, and $1-δ$ is the required confidence level. The method can be implemented on a classical computer and runs in time $O(M^4 \log(M/ε))$. …
