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1. CAFQA: A Classical Simulation Bootstrap for Variational Quantum Algorithms

   https://doi.org/10.1145/3567955.3567958  

   Ravi, Gokul Subramanian; Gokhale, Pranav; Ding, Yi; Kirby, William; Smith, Kaitlin; Baker, Jonathan M.; Love, Peter J.; Hoffmann, Henry; Brown, Kenneth R.; Chong, Frederic T. (December 2022, Proceedings of the 28th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 1 (ASPLOS 2023))

   Classical computing plays a critical role in the advancement of quantum frontiers in the NISQ era. In this spirit, this work uses classical simulation to bootstrap Variational Quantum Algorithms (VQAs). VQAs rely upon the iterative optimization of a parameterized unitary circuit (ansatz) with respect to an objective function. Since quantum machines are noisy and expensive resources, it is imperative to classically choose the VQA ansatz initial parameters to be as close to optimal as possible to improve VQA accuracy and accelerate their convergence on today’s devices. This work tackles the problem of finding a good ansatz initialization, by proposing CAFQA, a Clifford Ansatz For Quantum Accuracy. The CAFQA ansatz is a hardware-efficient circuit built with only Clifford gates. In this ansatz, the parameters for the tunable gates are chosen by searching efficiently through the Clifford parameter space via classical simulation. The resulting initial states always equal or outperform traditional classical initialization (e.g., Hartree-Fock), and enable high-accuracy VQA estimations. CAFQA is well-suited to classical computation because: a) Clifford-only quantum circuits can be exactly simulated classically in polynomial time, and b) the discrete Clifford space is searched efficiently via Bayesian Optimization. For the Variational Quantum Eigensolver (VQE) task of molecular ground state energy estimation (up to 18 qubits), CAFQA’s Clifford Ansatz achieves a mean accuracy of nearly 99% and recovers as much as 99.99% of the molecular correlation energy that is lost in Hartree-Fock initialization. CAFQA achieves mean accuracy improvements of 6.4x and 56.8x, over the state-of-the-art, on different metrics. The scalability of the approach allows for preliminary ground state energy estimation of the challenging chromium dimer (Cr2) molecule. With CAFQA’s high-accuracy initialization, the convergence of VQAs is shown to accelerate by 2.5x, even for small molecules. Furthermore, preliminary exploration of allowing a limited number of non-Clifford (T) gates in the CAFQA framework, shows that as much as 99.9% of the correlation energy can be recovered at bond lengths for which Clifford-only CAFQA accuracy is relatively limited, while remaining classically simulable. 

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2. CAFQA: Clifford Ansatz For Quantum Accuracy

   Gokul Subramanian Ravi, Pranav Gokhale (January 2022, ArXivorg)

   Variational Quantum Algorithms (VQAs) rely upon the iterative optimization of a parameterized unitary circuit with respect to an objective function. Since quantum machines are noisy and expensive resources, it is imperative to choose a VQA's ansatz appropriately and its initial parameters to be close to optimal. This work tackles the problem of finding initial ansatz parameters by proposing CAFQA, a Clifford ansatz for quantum accuracy. The CAFQA ansatz is a hardware-efficient circuit built with only Clifford gates. In this ansatz, the initial parameters for the tunable gates are chosen by searching efficiently through the Clifford parameter space via classical simulation, thereby producing a suitable stabilizer state. The stabilizer states produced are shown to always equal or outperform traditional classical initialization (e.g., Hartree-Fock), and often produce high accuracy estimations prior to quantum exploration. Furthermore, the technique is classically suited since a) Clifford circuits can be exactly simulated classically in polynomial time and b) the discrete Clifford space, while scaling exponentially in the number of qubits, is searched efficiently via Bayesian Optimization. For the Variational Quantum Eigensolver (VQE) task of molecular ground state energy estimation up to 20 qubits, CAFQA's Clifford Ansatz achieves a mean accuracy of near 99%, recovering as much as 99.99% of the correlation energy over Hartree-Fock. Notably, the scalability of the approach allows for preliminary ground state energy estimation of the challenging Chromium dimer with an accuracy greater than Hartree-Fock. With CAFQA's initialization, VQA convergence is accelerated by a factor of 2.5x. In all, this work shows that stabilizer states are an accurate ansatz initialization for VQAs. Furthermore, it highlights the potential for quantum-inspired classical techniques to support VQAs. 

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3. Synthesis of Logical Clifford Operators via Symplectic Geometry

   Rengaswamy, Narayanan; Calderbank, Robert; Kadhe, Swanand; Pfister, Henry D. (January 2018, IEEE International Symposium on Information Theory)

   Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in CN ×N as a 2m × 2m binary sym- plectic matrix, where N = 2m. We show that for an [m, m − k] stabilizer code every logical Clifford operator has 2k(k+1)/2 symplectic solutions, and we enumerate them efficiently using symplectic transvections. The desired circuits are then obtained by writing each of the solutions as a product of elementary symplectic matrices. For a given operator, our assembly of all of its physical realizations enables optimization over them with respect to a suitable metric. Our method of circuit synthesis can be applied to any stabilizer code, and this paper provides a proof of concept synthesis of universal Clifford gates for the well- known [6, 4, 2] code. Programs implementing our algorithms can be found at https://github.com/nrenga/symplectic-arxiv18a. 

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4. A quantum hamiltonian simulation benchmark

   https://doi.org/10.1038/s41534-022-00636-x  

   Dong, Yulong; Whaley, K. Birgitta; Lin, Lin (December 2022, npj Quantum Information)

   Abstract Hamiltonian simulation is one of the most important problems in quantum computation, and quantum singular value transformation (QSVT) is an efficient way to simulate a general class of Hamiltonians. However, the QSVT circuit typically involves multiple ancilla qubits and multi-qubit control gates. In order to simulate a certain class of n -qubit random Hamiltonians, we propose a drastically simplified quantum circuit that we refer to as the minimal QSVT circuit, which uses only one ancilla qubit and no multi-qubit controlled gates. We formulate a simple metric called the quantum unitary evolution score (QUES), which is a scalable quantum benchmark and can be verified without any need for classical computation. Under the globally depolarized noise model, we demonstrate that QUES is directly related to the circuit fidelity, and the potential classical hardness of an associated quantum circuit sampling problem. Under the same assumption, theoretical analysis suggests there exists an ‘optimal’ simulation time t opt  ≈ 4.81, at which even a noisy quantum device may be sufficient to demonstrate the potential classical hardness. 

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5. Linear growth of quantum circuit complexity

   https://doi.org/10.1038/s41567-022-01539-6  

   Haferkamp, Jonas; Faist, Philippe; Kothakonda, Naga B.; Eisert, Jens; Yunger Halpern, Nicole (May 2022, Nature Physics)

   Abstract The complexity of quantum states has become a key quantity of interest across various subfields of physics, from quantum computing to the theory of black holes. The evolution of generic quantum systems can be modelled by considering a collection of qubits subjected to sequences of random unitary gates. Here we investigate how the complexity of these random quantum circuits increases by considering how to construct a unitary operation from Haar-random two-qubit quantum gates. Implementing the unitary operation exactly requires a minimal number of gates—this is the operation’s exact circuit complexity. We prove a conjecture that this complexity grows linearly, before saturating when the number of applied gates reaches a threshold that grows exponentially with the number of qubits. Our proof overcomes difficulties in establishing lower bounds for the exact circuit complexity by combining differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits. 

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