Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Conference on VLSI Design

Optimized QAOA ansatz circuit design for two-body Hamiltonian problems

Ritajit Majumdar, Dhiraj Madan, Debasmita Bhoumik, Dhinakaran Vinayagamurthy, Shesha Raghunathan, Susmita Sur‐Kolay

Published Jan 6, 2024• 5 citations
Publisher

Abstract

Quantum Approximate Optimization Algorithm (QAOA) is a prospective candidate for providing quantum advantage in finding approximate solutions to optimization problems using near-term quantum devices. In this paper, the goal is to reduce the number of CNOT gates and the depth of quantum ansatz circuits for QAOA. First, we present a generalized QAOA formulation for any Hamiltonian that involves upto two-body interactions, as a graph. The circuit realization of the depth-p QAOA requires $2mp$ CNOT gates where the graph for the Hamiltonian has n vertices and m edges. Presently, a CNOT gate is one of the primary sources of error. We propose a graph-theoretic algorithm which can eliminate $n-1$ CNOTs from the QAOA ansatz while retaining functional equivalence with the original circuit. This improves upon previously proposed Depth First Search (DFS) method by restricting the increase in depth of the circuit, which was a drawback of the method, by $\simeq 84.8\%$ for Erdős-Renyi graphs. Finally, if the underlying connectivity (hardware coupling map) and the initial qubit placement for a hardware are known, then we present how to extend the greedy heuristic method in order to obtain a functionally equivalent circuit with $\simeq 5\%$; fewer SWAP gates for Erdős-Renyi graphs.

DOI
10.1109/vlsid60093.2024.00072

Related papers

Optimizing Ansatz Design in QAOA for Max-cut

Ritajit Majumdar, Dhiraj Madan, Debasmita Bhoumik +3 more

Quantum Approximate Optimization Algorithm (QAOA) is studied primarily to find approximate solutions to combinatorial optimization problems. For a graph with $n$ vertices and $m$ edges, a depth $p$ QAOA for the Max-cut problem requires $2\cdot m \cdot p$ CNOT gates. CNOT is one of the primary sources of error in modern quantum computers. In this paper, we propose two hardware independent methods to reduce the number of CNOT gates in the circuit. First, we present a method based on Edge Coloring of the input graph that minimizes the the number of cycles (termed as depth of the circuit), and reduces upto $\lfloor \frac{n}{2} \rfloor$ CNOT gates. Next, we depict another method based on Depth First Search (DFS) on the input graph that reduces $n-1$ CNOT gates, but increases depth of the circuit moderately. We analytically derive the condition for which the reduction in CNOT gates overshadows this increase in depth, and the error probability of the circuit is still lowered. We show that all IBM Quantum Hardware satisfy this condition. We simulate these two methods for graphs of various sparsity with the \textit{ibmq\_manhattan} noise model, and show that the DFS based method outperforms the edge coloring based method, which in turn, outperforms the traditional QAOA circuit in terms of reduction in the number of CNOT gates, and hence the probability of error of the circuit.

arXiv:2106.02812•Jun 5, 2021•18 citations

Depth Optimized Ansatz Circuit in QAOA for Max-Cut

Ritajit Majumdar, Debasmita Bhoumik, Dhiraj Madan +3 more

While a Quantum Approximate Optimization Algorithm (QAOA) is intended to provide a quantum advantage in finding approximate solutions to combinatorial optimization problems, noise in the system is a hurdle in exploiting its full potential. Several error mitigation techniques have been studied to lessen the effect of noise on this algorithm. Recently, Majumdar et al. proposed a Depth First Search (DFS) based method to reduce $n-1$ CNOT gates in the ansatz design of QAOA for finding Max-Cut in a graph G = (V, E), |V| = n. However, this method tends to increase the depth of the circuit, making it more prone to relaxation error. The depth of the circuit is proportional to the height of the DFS tree, which can be $n-1$ in the worst case. In this paper, we propose an $O(Δ\cdot n^2)$ greedy heuristic algorithm, where $Δ$ is the maximum degree of the graph, that finds a spanning tree of lower height, thus reducing the overall depth of the circuit while still retaining the $n-1$ reduction in the number of CNOT gates needed in the ansatz. We numerically show that this algorithm achieves nearly 10 times increase in the probability of success for each iteration of QAOA for Max-Cut. We further show that although the average depth of the circuit produced by this heuristic algorithm still grows linearly with n, our algorithm reduces the slope of the linear increase from 1 to 0.11.

arXiv:2110.04637•Oct 9, 2021•3 citations

Improved Qubit Routing for QAOA Circuits

Ayse Kotil, Fedor Šimkovic, Martin Leib

We develop a qubit routing algorithm with polynomial classical run time for the Quantum Approximate Optimization Algorithm (QAOA). The algorithm follows a two step process. First, it obtains a near-optimal solution, based on Vizing's theorem for the edge coloring problem, consisting of subsets of the interaction gates that can be executed in parallel on a fully parallelized all-to-all connected QPU. Second, it proceeds with greedy application of SWAP gates based on their net effect on the distance of remaining interaction gates on a specific hardware connectivity graph. Our algorithm strikes a balance between optimizing for both the circuit depth and total SWAP gate count. We show that it improves upon existing state-of-the-art routing algorithms for QAOA circuits defined on $k$-regular as well as Erdös-Renyi problem graphs of sizes up to $N \leq 400$.

arXiv:2312.15982•Dec 26, 2023•1 citation

Multi-angle quantum approximate optimization algorithm

Rebekah Herrman, Phillip C. Lotshaw, James Ostrowski +2 more

The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we investigate a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time for a test dataset we consider. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.

arXiv:2109.11455•Scientific Reports•Apr 26, 2022•117 citations

Scaling quantum approximate optimization on near-term hardware

Phillip C. Lotshaw, Thien Nguyen, Anthony Santana +5 more

The quantum approximate optimization algorithm (QAOA) is an approach for near-term quantum computers to potentially demonstrate computational advantage in solving combinatorial optimization problems. However, the viability of the QAOA depends on how its performance and resource requirements scale with problem size and complexity for realistic hardware implementations. Here, we quantify scaling of the expected resource requirements by synthesizing optimized circuits for hardware architectures with varying levels of connectivity. Assuming noisy gate operations, we estimate the number of measurements needed to sample the output of the idealized QAOA circuit with high probability. We show the number of measurements, and hence total time to solution, grows exponentially in problem size and problem graph degree as well as depth of the QAOA ansatz, gate infidelities, and inverse hardware graph degree. These problems may be alleviated by increasing hardware connectivity or by recently proposed modifications to the QAOA that achieve higher performance with fewer circuit layers.

arXiv:2201.02247•Scientific Reports•Jul 20, 2022•37 citations

Shortcuts to the quantum approximate optimization algorithm

Yahui Chai, Yong‐Jian Han, Yu-Chun Wu +3 more

The quantum approximate optimization algorithm (QAOA) is a quantum-classical hybrid algorithm intending to find the ground state of a target Hamiltonian. Theoretically, QAOA can obtain the approximate solution if the quantum circuit is deep enough. Actually, the performance of QAOA decreases practically if the quantum circuit is deep since near-term devices are not noise free, and the errors caused by noise accumulate as the quantum circuit increases. In order to reduce the depth of quantum circuits, we propose an ansatz dubbed as ``Shortcuts to QAOA'' (S-QAOA), S-QAOA provides shortcuts to the ground state of the target Hamiltonian by including more two-body interactions and releasing the parameter freedoms. To be specific, besides the existing $ZZ$ interaction in the QAOA ansatz, other two-body interactions are introduced in the S-QAOA ansatz such that the approximate solutions could be obtained with smaller circuit depth. Considering the Max-Cut problem and the Sherrington-Kirkpatrick model, numerically computation shows the $YY$ interaction has the best performance. The reason for this might arise from the counterdiabatic effect generated by $YY$ interaction. On top of this, we release the freedom of parameters of two-body interactions, which a priori do not necessarily have to be fully identical, and numerical results show that it is worth paying the extra cost of having more parameter freedom since one has a greater improvement on success rate.

arXiv:2112.10943•Physical Review A•Apr 8, 2022•20 citations
Quantum Approximate Optimization Algorithm (QAOA) is a prospective candidate for providing quantum advantage in finding approximate solutions to optimization problems using near-term quantum devices. In this paper, the goal is to reduce the number of CNOT gates and the depth of quantum ansatz circuits for QAOA. First, we present a generalized QAOA formulation for any Hamiltonian that involves upto two-body interactions, as a graph. The circuit realization of the depth-p QAOA requires 2mp2mp2mp CNOT gates where the graph for the Hamiltonian has n vertices and m edges. Presently, a CNOT gate is one of the primary sources of error. We propose a graph-theoretic algorithm which can eliminate n−1n-1n−1 CNOTs from the QAOA ansatz while retaining functional equivalence with the original circuit. This improves upon previously proposed Depth First Search (DFS) method by restricting the increase in depth of the circuit, which was a drawback of the method, by ≃84.8%\simeq 84.8\%≃84.8% for Erdős-Renyi graphs. Finally, if the underlying connectivity (hardware coupling map) and the initial qubit placement for a hardware are known, then we present how to extend the greedy heuristic method in order to obtain a functionally equivalent circuit with ≃5%\simeq 5\%≃5%; fewer SWAP gates for Erdős-Renyi graphs.
Quantum Approximate Optimization Algorithm (QAOA) is studied primarily to find approximate solutions to combinatorial optimization problems. For a graph with nnn vertices and mmm edges, a depth ppp QAOA for the Max-cut problem requires 2⋅m⋅p2\cdot m \cdot p2⋅m⋅p CNOT gates. CNOT is one of the primary sources of error in modern quantum computers. In this paper, we propose two hardware independent methods to reduce the number of CNOT gates in the circuit. First, we present a method based on Edge Coloring of the input graph that minimizes the the number of cycles (termed as depth of the circuit), and reduces upto ⌊n2⌋\lfloor \frac{n}{2} \rfloor⌊2n​⌋ CNOT gates. Next, we depict another method based on Depth First Search (DFS) on the input graph that reduces n−1n-1n−1 CNOT gates, but increases depth of the circuit moderately. We analytically derive the condition for which the reduction in CNOT gates overshadows this increase in depth, and the error probability of the circuit is still lowered. We show that all IBM Quantum Hardware satisfy this condition. We simulate these two methods for graphs of various sparsity with the \textit{ibmq\_manhattan} noise model, and show that the DFS based method outperforms the edge coloring based method, which in turn, outperforms the traditional QAOA circuit in terms of reduction in the number of CNOT gates, and hence the probability of error of the circuit.
While a Quantum Approximate Optimization Algorithm (QAOA) is intended to provide a quantum advantage in finding approximate solutions to combinatorial optimization problems, noise in the system is a hurdle in exploiting its full potential. Several error mitigation techniques have been studied to lessen the effect of noise on this algorithm. Recently, Majumdar et al. proposed a Depth First Search (DFS) based method to reduce n−1n-1n−1 CNOT gates in the ansatz design of QAOA for finding Max-Cut in a graph G = (V, E), |V| = n. However, this method tends to increase the depth of the circuit, making it more prone to relaxation error. The depth of the circuit is proportional to the height of the DFS tree, which can be n−1n-1n−1 in the worst case. In this paper, we propose an O(Δ⋅n2)O(Δ\cdot n^2)O(Δ⋅n2) greedy heuristic algorithm, where ΔΔΔ is the maximum degree of the graph, that finds a spanning tree of lower height, thus reducing the overall depth of the circuit while still retaining the n−1n-1n−1 reduction in the number of CNOT gates needed in the ansatz. We numerically show that this algorithm achieves nearly 10 times increase in the probability of success for each iteration of QAOA for Max-Cut. We further show that although the average depth of the circuit produced by this heuristic algorithm still grows linearly with n, our algorithm reduces the slope of the linear increase from 1 to 0.11.
We develop a qubit routing algorithm with polynomial classical run time for the Quantum Approximate Optimization Algorithm (QAOA). The algorithm follows a two step process. First, it obtains a near-optimal solution, based on Vizing's theorem for the edge coloring problem, consisting of subsets of the interaction gates that can be executed in parallel on a fully parallelized all-to-all connected QPU. Second, it proceeds with greedy application of SWAP gates based on their net effect on the distance of remaining interaction gates on a specific hardware connectivity graph. Our algorithm strikes a balance between optimizing for both the circuit depth and total SWAP gate count. We show that it improves upon existing state-of-the-art routing algorithms for QAOA circuits defined on kkk-regular as well as Erdös-Renyi problem graphs of sizes up to N≤400N \leq 400N≤400.
The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we investigate a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time for a test dataset we consider. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.
The quantum approximate optimization algorithm (QAOA) is an approach for near-term quantum computers to potentially demonstrate computational advantage in solving combinatorial optimization problems. However, the viability of the QAOA depends on how its performance and resource requirements scale with problem size and complexity for realistic hardware implementations. Here, we quantify scaling of the expected resource requirements by synthesizing optimized circuits for hardware architectures with varying levels of connectivity. Assuming noisy gate operations, we estimate the number of measurements needed to sample the output of the idealized QAOA circuit with high probability. We show the number of measurements, and hence total time to solution, grows exponentially in problem size and problem graph degree as well as depth of the QAOA ansatz, gate infidelities, and inverse hardware graph degree. These problems may be alleviated by increasing hardware connectivity or by recently proposed modifications to the QAOA that achieve higher performance with fewer circuit layers.
The quantum approximate optimization algorithm (QAOA) is a quantum-classical hybrid algorithm intending to find the ground state of a target Hamiltonian. Theoretically, QAOA can obtain the approximate solution if the quantum circuit is deep enough. Actually, the performance of QAOA decreases practically if the quantum circuit is deep since near-term devices are not noise free, and the errors caused by noise accumulate as the quantum circuit increases. In order to reduce the depth of quantum circuits, we propose an ansatz dubbed as ``Shortcuts to QAOA'' (S-QAOA), S-QAOA provides shortcuts to the ground state of the target Hamiltonian by including more two-body interactions and releasing the parameter freedoms. To be specific, besides the existing ZZZZZZ interaction in the QAOA ansatz, other two-body interactions are introduced in the S-QAOA ansatz such that the approximate solutions could be obtained with smaller circuit depth. Considering the Max-Cut problem and the Sherrington-Kirkpatrick model, numerically computation shows the YYYYYY interaction has the best performance. The reason for this might arise from the counterdiabatic effect generated by YYYYYY interaction. On top of this, we release the freedom of parameters of two-body interactions, which a priori do not necessarily have to be fully identical, and numerical results show that it is worth paying the extra cost of having more parameter freedom since one has a greater improvement on success rate.