One of the main concerns in the era of noisy intermediate-scale quantum (NISQ) computing and fault-tolerant quantum computing is the optimization of circuit implementation for quantum oracles, particularly with limited resources. Synthesizing a satisfiability (SAT) oracle, a crucial component in solving SAT problems, presents a significant challenge. The current state-of-the-art implementation of an$m$-clause SAT-oracle necessitates$2m-1$ancillary qubits and a linear number of elementary gates. We develop two efficient and ancilla-adjustable synthesis algorithms to reduce the overall quantum resource usage. Our first quantum oracle algorithm achieves quadratic optimization in the number of ancillary qubits with merely eight times increased circuit size. We also show that using only three ancillary qubits with quadratic circuit size expansion is enough. Our second algorithm optimizes the circuit depth of the SAT oracle to$\tilde {O}(\log m)$using$m$ancillary qubits. By running our algorithms on classical intractable SAT instances featured in SAT competitions, the experiment results show that our required quantum resources align well with our theoretical analysis. Our algorithms highlight the scalability of SAT-oracle-based algorithms in near-term quantum devices, such as Grover’s algorithm.
One of the main concerns in the era of noisy intermediate-scale quantum (NISQ) computing and fault-tolerant quantum computing is the optimization of circuit implementation for quantum oracles, particularly with limited resources. Synthesizing a satisfiability (SAT) oracle, a crucial component in solving SAT problems, presents a significant challenge. The current state-of-the-art implementation of anm-clause SAT-oracle necessitates2m−1ancillary qubits and a linear number of elementary gates. We develop two efficient and ancilla-adjustable synthesis algorithms to reduce the overall quantum resource usage. Our first quantum oracle algorithm achieves quadratic optimization in the number of ancillary qubits with merely eight times increased circuit size. We also show that using only three ancillary qubits with quadratic circuit size expansion is enough. Our second algorithm optimizes the circuit depth of the SAT oracle toO~(logm)usingmancillary qubits. By running our algorithms on classical intractable SAT instances featured in SAT competitions, the experiment results show that our required quantum resources align well with our theoretical analysis. Our algorithms highlight the scalability of SAT-oracle-based algorithms in near-term quantum devices, such as Grover’s algorithm.
To demonstrate the advantage of quantum computation, many attempts have been made to attack classically intractable problems, such as the satisfiability problem (SAT), with quantum computer. To use quantum algorithms to solve these NP-hard problems, a quantum oracle with quantum circuit implementation is usually required. In this manuscript, we first introduce a novel algorithm to synthesize quantum logic in the Conjunctive Normal Form (CNF) model. Compared with linear ancillary qubits in the implementation of Qiskit open-source framework, our algorithm can synthesize an m clauses n variables k-CNF with O(k2m2/n) quantum gates by only using three ancillary qubits. Both the size and depth of the circuit can be further compressed with the increase in the number of ancillary qubits. When the number of ancillary qubits is Ω(mϵ) (for any ϵ>0), the size of the quantum circuit given by the algorithm is O(km), which is asymptotically optimal. Furthermore, we design another algorithm to optimize the depth of the quantum circuit with only a small increase in the size of the quantum circuit. Experiments show that our algorithms have great improvement in size and depth compared with the previous algorithms.
Several prominent quantum computing algorithms—including Grover’s search algorithm and Shor’s algorithm for finding the prime factorization of an integer—employ subcircuits termed ‘oracles’ that embed a specific instance of a mathematical function into a corresponding bijective function that is then realized as a quantum circuit representation. Designing oracles, and particularly, designing them to be optimized for a particular use case, can be a non-trivial task. For example, the challenge of implementing quantum circuits in the current era of NISQ-based quantum computers generally dictates that they should be designed with a minimal number of qubits, as larger qubit counts increase the likelihood that computations will fail due to one or more of the qubits decohering. However, some quantum circuits require that function domain values be preserved, which can preclude using the minimal number of qubits in the oracle circuit. Thus, quantum oracles must be designed with a particular application in mind. In this work, we present two methods for automatic quantum oracle synthesis. One of these methods uses a minimal number of qubits, while the other preserves the function domain values while also minimizing the overall required number of qubits. For each method, we describe known quantum circuit use cases, and illustrate implementation using an automated quantum compilation and optimization tool to synthesize oracles for a set of benchmark functions; we can then compare the methods with metrics including required qubit count and quantum circuit complexity.
The current phase of quantum computing is in the Noisy Intermediate-Scale Quantum (NISQ) era. On NISQ devices, two-qubit gates such as CNOTs are much noisier than single-qubit gates, so it is essential to minimize their count. Quantum circuit synthesis is a process of decomposing an arbitrary unitary into a sequence of quantum gates, and can be used as an optimization tool to produce shorter circuits to improve overall circuit fidelity. However, the time-to-solution of synthesis grows exponentially with the number of qubits. As a result, synthesis is intractable for circuits on a large qubit scale. In this paper, we propose a hierarchical, block-by-block optimization framework, QGo, for quantum circuit optimization. Our approach allows an exponential cost optimization to scale to large circuits. QGo uses a combination of partitioning and synthesis: 1) partition the circuit into a sequence of independent circuit blocks; 2) re-generate and optimize each block using quantum synthesis; and 3) re-compose the final circuit by stitching all the blocks together. We perform our analysis and show the fidelity improvements in three different regimes: small-size circuits on real devices, medium-size circuits on noise simulations, and large-size circuits on analytical models. Using a set of NISQ benchmarks, we show that QGo can reduce the number of CNOT gates by 29.9% on average and up to 50% when compared with industrial compilers such as t|ket>. When executed on the IBM Athens system, shorter depth leads to higher circuit fidelity. We also demonstrate the scalability of our QGo technique to optimize circuits of 60+ qubits. Our technique is the first demonstration of successfully employing and scaling synthesis in the compilation toolchain for large circuits. Overall, our approach is robust for direct incorporation in production compiler toolchains.
Quantum oracles play a fundamental role in many quantum algorithms, such as Grover's search and quantum counting. In oracle synthesis, reducing the circuit depth under a limited number of ancilla qubits is essential, given the constraints of current quantum devices. However, for satisfiability oracles (SAT-oracles) under a limited ancilla budget, existing synthesis methods fail to fully exploit clause-level parallelism, leading to deep circuits. In practice, evaluating clauses in parallel can quickly exhaust the ancilla budget. To address this issue, we introduce the Clustered Synthesis Tree (CST), a depth-oriented SAT-oracle synthesis framework that exploits clause-level parallelism while respecting the ancilla budget. CST leverages a recursive synthesis tree as the backbone for scheduling clause evaluations and groups its clause leaves into clusters, whose clauses are evaluated in parallel by a reversible oracle called ClausePack. To form the clusters, CST introduces a polynomial-time heuristic called SeedGrow for the NP-hard clause-grouping problem, and the resulting tree is compiled into an executable circuit by CST-Map. Compared with the depth-oriented pAND baseline, CST can reduce the circuit depth by an average of 10.2x and 8.8x-11.0x on SATLIB and random 4-CNF benchmarks, respectively, under the same ancilla budgets, while also outperforming the width-oriented pAND, clause-storing, and qSAT baselines.
The current phase of quantum computing is in the Noisy Intermediate-Scale Quantum (NISQ) era. On NISQ devices, two-qubit gates such as CNOTs are much noisier than single-qubit gates, so it is essential to minimize their count. Quantum circuit synthesis is a process of decomposing an arbitrary unitary into a sequence of quantum gates, and can be used as an optimization tool to produce shorter circuits to improve overall circuit fidelity. However, the time-to-solution of synthesis grows exponentially with the number of qubits. As a result, synthesis is intractable for circuits on a large qubit scale.
In this paper, we propose a hierarchical, block-by-block optimization framework, QGo, for quantum circuit optimization. Our approach allows an exponential cost optimization to scale to large circuits. QGo uses a combination of partitioning and synthesis: 1) partition the circuit into a sequence of independent circuit blocks; 2) re-generate and optimize each block using quantum synthesis; and 3) re-compose the final circuit by stitching all the blocks together. We perform our analysis and show the fidelity improvements in three different regimes: small-size circuits on real devices, medium-size circuits on noise simulations, and large-size circuits on analytical models. Using a set of NISQ benchmarks, we show that QGo can reduce the number of CNOT gates by 29.9% on average and up to 50% when compared with industrial compilers such as t|ket>. When executed on the IBM Athens system, shorter depth leads to higher circuit fidelity. We also demonstrate the scalability of our QGo technique to optimize circuits of 60+ qubits. Our technique is the first demonstration of successfully employing and scaling synthesis in the compilation toolchain for large circuits. Overall, our approach is robust for direct incorporation in production compiler toolchains.
Current quantum devices have unutilized high-level quantum resources. More and more attention has been paid to the qudit quantum systems with larger than two dimensions to maximize the potential computing power of quantum computation. Then, a natural problem arises: How do we implement quantum algorithms on qudit quantum systems? In this work, we propose a novel qudit phase gadget method for synthesizing the qudit diagonal unitary matrices. This method is suitable for the Noisy Intermediate-Scale Quantum (NISQ) and fault-tolerant eras due to its versatility in different connectivity architectures and the optimality of its resource consumption. The method can work on any connectivity architecture with asymptotic optimal circuit depth and size. For a 10-qutrit diagonal unitary, our algorithm reduces the circuit depth form about 100000 to 500 with 300 ancillary qutrits. Further, this method can be promoted to different quantum circuit synthesis problems, such as quantum state preparation problems, general unitary synthesis problems, etc.