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1. Logical abstractions for noisy variational Quantum algorithm simulation

   https://doi.org/10.1145/3445814.3446750  

   Huang, Yipeng ; Holtzen, Steven ; Millstein, Todd ; Van den Broeck, Guy ; Martonosi, Margaret ( April 2021 , 26th ACM International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS ’21))

   null (Ed.)

   Due to the unreliability and limited capacity of existing quantum computer prototypes, quantum circuit simulation continues to be a vital tool for validating next generation quantum computers and for studying variational quantum algorithms, which are among the leading candidates for useful quantum computation. Existing quantum circuit simulators do not address the common traits of variational algorithms, namely: 1) their ability to work with noisy qubits and operations, 2) their repeated execution of the same circuits but with different parameters, and 3) the fact that they sample from circuit final wavefunctions to drive a classical optimization routine. We present a quantum circuit simulation toolchain based on logical abstractions targeted for simulating variational algorithms. Our proposed toolchain encodes quantum amplitudes and noise probabilities in a probabilistic graphical model, and it compiles the circuits to logical formulas that support efficient repeated simulation of and sampling from quantum circuits for different parameters. Compared to state-of-the-art state vector and density matrix quantum circuit simulators, our simulation approach offers greater performance when sampling from noisy circuits with at least eight to 20 qubits and with around 12 operations on each qubit, making the approach ideal for simulating near-term variational quantum algorithms. And for simulating noise-free shallow quantum circuits with 32 qubits, our simulation approach offers a 66X reduction in sampling cost versus quantum circuit simulation techniques based on tensor network contraction. 

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2. Efficient classical simulation of noisy random quantum circuits in one dimension

   https://doi.org/10.22331/q-2020-09-11-318  

   Noh, Kyungjoo ; Jiang, Liang ; Fefferman, Bill ( September 2020 , Quantum)

   Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers. 

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3. How many qubits are needed for quantum computational supremacy?

   https://doi.org/10.22331/q-2020-05-11-264  

   Dalzell, Alexander M. ; Harrow, Aram W. ; Koh, Dax Enshan ; La Placa, Rolando L. ( May 2020 , Quantum)

   null (Ed.)

   Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require some sort of computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy ( P H ) does not collapse, a stronger version of the statement that P ≠ N P , which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic nature of this conclusion prevents us from calculating exactly how many qubits these quantum circuits must have for their classical simulation to be intractable on modern classical supercomputers. We refine these quantum computational supremacy arguments and perform such a calculation by imposing fine-grained versions of the non-collapse conjecture. Our first two conjectures poly3-NSETH( a ) and per-int-NSETH( b ) take specific classical counting problems related to the number of zeros of a degree-3 polynomial in n variables over F 2 or the permanent of an n × n integer-valued matrix, and assert that any non-deterministic algorithm that solves them requires 2 c n time steps, where c ∈ { a , b } . A third conjecture poly3-ave-SBSETH( a ′ ) asserts a similar statement about average-case algorithms living in the exponential-time version of the complexity class S B P . We analyze evidence for these conjectures and argue that they are plausible when a = 1 / 2 , b = 0.999 and a ′ = 1 / 2 .Imposing poly3-NSETH(1/2) and per-int-NSETH(0.999), and assuming that the runtime of a hypothetical quantum circuit simulation algorithm would scale linearly with the number of gates/constraints/optical elements, we conclude that Instantaneous Quantum Polynomial-Time (IQP) circuits with 208 qubits and 500 gates, Quantum Approximate Optimization Algorithm (QAOA) circuits with 420 qubits and 500 constraints and boson sampling circuits (i.e. linear optical networks) with 98 photons and 500 optical elements are large enough for the task of producing samples from their output distributions up to constant multiplicative error to be intractable on current technology. Imposing poly3-ave-SBSETH(1/2), we additionally rule out simulations with constant additive error for IQP and QAOA circuits of the same size. Without the assumption of linearly increasing simulation time, we can make analogous statements for circuits with slightly fewer qubits but requiring 10 4 to 10 7 gates. 

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4. Avoiding symmetry roadblocks and minimizing the measurement overhead of adaptive variational quantum eigensolvers

   https://doi.org/10.22331/q-2023-06-12-1040  

   Shkolnikov, V. O. ; Mayhall, Nicholas J. ; Economou, Sophia E. ; Barnes, Edwin ( June 2023 , Quantum)

   Quantum simulation of strongly correlated systems is potentially the most feasible useful application of near-term quantum computers. Minimizing quantum computational resources is crucial to achieving this goal. A promising class of algorithms for this purpose consists of variational quantum eigensolvers (VQEs). Among these, problem-tailored versions such as ADAPT-VQE that build variational ansätze step by step from a predefined operator pool perform particularly well in terms of circuit depths and variational parameter counts. However, this improved performance comes at the expense of an additional measurement overhead compared to standard VQEs. Here, we show that this overhead can be reduced to an amount that grows only linearly with the number$n$of qubits, instead of quartically as in the original ADAPT-VQE. We do this by proving that operator pools of size$2n-2$can represent any state in Hilbert space if chosen appropriately. We prove that this is the minimal size of such complete pools, discuss their algebraic properties, and present necessary and sufficient conditions for their completeness that allow us to find such pools efficiently. We further show that, if the simulated problem possesses symmetries, then complete pools can fail to yield convergent results, unless the pool is chosen to obey certain symmetry rules. We demonstrate the performance of such symmetry-adapted complete pools by using them in classical simulations of ADAPT-VQE for several strongly correlated molecules. Our findings are relevant for any VQE that uses an ansatz based on Pauli strings.

    

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5. Entangled quantum cellular automata, physical complexity, and Goldilocks rules

   Logan E. Hillberry, Matthew T. ( May 2020 , ArXivorg)

   Cellular automata are interacting classical bits that display diverse behaviors, from fractals to random-number generators to Turing-complete computation. We introduce entangled quantum cellular automata subject to Goldilocks rules, tradeoffs of the kind underpinning biological, social, and economic complexity. Tweaking digital and analog quantum-computing protocols generates persistent entropy fluctuations; robust dynamical features, including an entangled breather; and network structure and dynamics consistent with complexity. Present-day quantum platforms---Rydberg arrays, trapped ions, and superconducting qubits---can implement Goldilocks protocols, which generate quantum many-body states with rich entanglement and structure. Moreover, the complexity studies reported here underscore an emerging idea in many-body quantum physics: some systems fall outside the integrable/chaotic dichotomy. 

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