scholarly journals Quantum advantage from energy measurements of many-body quantum systems

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 465
Author(s):  
Leonardo Novo ◽  
Juani Bermejo-Vega ◽  
Raúl García-Patrón
Keyword(s):  
High Resolution ◽  
Measurement Errors ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Polynomial Hierarchy ◽  
Many Body ◽  
Classical Simulation ◽  
Measurement Resolution ◽  
New Perspective ◽  
Sampling Problems

The problem of sampling outputs of quantum circuits has been proposed as a candidate for demonstrating a quantum computational advantage (sometimes referred to as quantum "supremacy"). In this work, we investigate whether quantum advantage demonstrations can be achieved for more physically-motivated sampling problems, related to measurements of physical observables. We focus on the problem of sampling the outcomes of an energy measurement, performed on a simple-to-prepare product quantum state – a problem we refer to as energy sampling. For different regimes of measurement resolution and measurement errors, we provide complexity theoretic arguments showing that the existence of efficient classical algorithms for energy sampling is unlikely. In particular, we describe a family of Hamiltonians with nearest-neighbour interactions on a 2D lattice that can be efficiently measured with high resolution using a quantum circuit of commuting gates (IQP circuit), whereas an efficient classical simulation of this process should be impossible. In this high resolution regime, which can only be achieved for Hamiltonians that can be exponentially fast-forwarded, it is possible to use current theoretical tools tying quantum advantage statements to a polynomial-hierarchy collapse whereas for lower resolution measurements such arguments fail. Nevertheless, we show that efficient classical algorithms for low-resolution energy sampling can still be ruled out if we assume that quantum computers are strictly more powerful than classical ones. We believe our work brings a new perspective to the problem of demonstrating quantum advantage and leads to interesting new questions in Hamiltonian complexity.

scholarly journals How many qubits are needed for quantum computational supremacy?

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 264 ◽  
Author(s):  
Alexander M. Dalzell ◽  
Aram W. Harrow ◽  
Dax Enshan Koh ◽  
Rolando L. La Placa
Keyword(s):  
Optical Networks ◽  
Quantum Computer ◽  
Circuit Simulation ◽  
Quantum Circuit ◽  
Deterministic Algorithm ◽  
Quantum Circuits ◽  
Polynomial Hierarchy ◽  
Optical Elements ◽  
Average Case ◽  
Classical Simulation

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 (PH) does not collapse, a stronger version of the statement that P≠NP, 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 F2 or the permanent of an n×n integer-valued matrix, and assert that any non-deterministic algorithm that solves them requires 2cn 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 SBP. 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 104 to 107 gates.

scholarly journals Hyper-optimized tensor network contraction

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 410
Author(s):  
Johnnie Gray ◽  
Stefanos Kourtis
Keyword(s):  
Computation Time ◽  
Quantum Circuit ◽  
Optimization Approach ◽  
Many Body ◽  
Tensor Networks ◽  
Randomized Protocols ◽  
Classical Simulation ◽  
Tensor Network ◽  
Speed Up ◽  
Many Body Systems

Tensor networks represent the state-of-the-art in computational methods across many disciplines, including the classical simulation of quantum many-body systems and quantum circuits. Several applications of current interest give rise to tensor networks with irregular geometries. Finding the best possible contraction path for such networks is a central problem, with an exponential effect on computation time and memory footprint. In this work, we implement new randomized protocols that find very high quality contraction paths for arbitrary and large tensor networks. We test our methods on a variety of benchmarks, including the random quantum circuit instances recently implemented on Google quantum chips. We find that the paths obtained can be very close to optimal, and often many orders or magnitude better than the most established approaches. As different underlying geometries suit different methods, we also introduce a hyper-optimization approach, where both the method applied and its algorithmic parameters are tuned during the path finding. The increase in quality of contraction schemes found has significant practical implications for the simulation of quantum many-body systems and particularly for the benchmarking of new quantum chips. Concretely, we estimate a speed-up of over 10,000× compared to the original expectation for the classical simulation of the Sycamore `supremacy' circuits.

scholarly journals Further extensions of Clifford circuits and their classical simulation complexities

2017 ◽  
Vol 17 (3&4) ◽  
pp. 262-282
Author(s):  
Dax E. Koh
Keyword(s):  
Quantum Circuit ◽  
Complete Classification ◽  
Polynomial Hierarchy ◽  
New Combinations ◽  
Computational Power ◽  
Classical Simulation

Extended Clifford circuits straddle the boundary between classical and quantum computational power. Whether such circuits are efficiently classically simulable seems to depend delicately on the ingredients of the circuits. While some combinations of ingredients lead to efficiently classically simulable circuits, other combinations, which might just be slightly different, lead to circuits which are likely not. We extend the results of Jozsa and Van den Nest [Quant. Info. Comput. 14, 633 (2014)] by studying two further extensions of Clifford circuits. First, we consider how the classical simulation complexity changes when we allow for more general measurements. Second, we investigate different notions of what it means to ‘classically simulate’ a quantum circuit. These further extensions give us 24 new combinations of ingredients compared to Jozsa and Van den Nest, and we give a complete classification of their classical simulation complexities. Our results provide more examples where seemingly modest changes to the ingredients of Clifford circuits lead to “large” changes in the classical simulation complexities of the circuits, and also include new examples of extended Clifford circuits that exhibit “quantum supremacy”, in the sense that it is not possible to efficiently classically sample from the output distributions of such circuits, unless the polynomial hierarchy collapses.

scholarly journals Clifford recompilation for faster classical simulation of quantum circuits

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 170
Author(s):  
Hammam Qassim ◽  
Joel J. Wallman ◽  
Joseph Emerson
Keyword(s):  
Fault Tolerant ◽  
State Of The Art ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Monte Carlo Algorithm ◽  
Quantum Devices ◽  
Output State ◽  
Classical Simulation ◽  
Main Technique ◽  
Near Term

Simulating quantum circuits classically is an important area of research in quantum information, with applications in computational complexity and validation of quantum devices. One of the state-of-the-art simulators, that of Bravyi et al, utilizes a randomized sparsification technique to approximate the output state of a quantum circuit by a stabilizer sum with a reduced number of terms. In this paper, we describe an improved Monte Carlo algorithm for performing randomized sparsification. This algorithm reduces the runtime of computing the approximate state by the factorℓ/m, whereℓandmare respectively the total and non-Clifford gate counts. The main technique is a circuit recompilation routine based on manipulating exponentiated Pauli operators. The recompilation routine also facilitates numerical search for Clifford decompositions of products of non-Clifford gates, which can further reduce the runtime in certain cases by reducing the 1-norm of the vector of expansion,‖a‖1. It may additionally lead to a framework for optimizing circuit implementations over a gate-set, reducing the overhead for state-injection in fault-tolerant implementations. We provide a concise exposition of randomized sparsification, and describe how to use it to estimate circuit amplitudes in a way which can be generalized to a broader class of gates and states. This latter method can be used to obtain additive error estimates of circuit probabilities with a faster runtime than the full techniques of Bravyi et al. Such estimates are useful for validating near-term quantum devices provided that the target probability is not exponentially small.

scholarly journals Exponential quantum speed-ups are generic

2013 ◽  
Vol 13 (11&12) ◽  
pp. 901-924
Author(s):  
Fernando G.S.L. Brandao ◽  
Michal Horodecki
Keyword(s):  
Quantum Circuit ◽  
Black Box ◽  
Quantum Circuits ◽  
Query Complexity ◽  
Many Body ◽  
Many Body Theory ◽  
Exponential Separation ◽  
Local Quantum ◽  
Bounded Error ◽  
Body Theory

A central problem in quantum computation is to understand which quantum circuits are useful for exponential speed-ups over classical computation. We address this question in the setting of query complexity and show that for almost any sufficiently long quantum circuit one can construct a black-box problem which is solved by the circuit with a constant number of quantum queries, but which requires exponentially many classical queries, even if the classical machine has the ability to postselect. We prove the result in two steps. In the first, we show that almost any element of an approximate unitary 3-design is useful to solve a certain black-box problem efficiently. The problem is based on a recent oracle construction of Aaronson and gives an exponential separation between quantum and classical post-selected bounded-error query complexities. In the second step, which may be of independent interest, we prove that linear-sized random quantum circuits give an approximate unitary 3-design. The key ingredient in the proof is a technique from quantum many-body theory to lower bound the spectral gap of local quantum Hamiltonians.

scholarly journals Commuting quantum circuits with few outputs are unlikely to be classically simulatable

2016 ◽  
Vol 16 (3&4) ◽  
pp. 251-270 ◽  
Author(s):  
Yasuhiro Takahashi ◽  
Seiichiro Tani ◽  
Takeshi Yamazaki ◽  
Kazuyuki Tanaka
Keyword(s):  
Probability Distribution ◽  
Polynomial Time ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Polynomial Hierarchy ◽  
The Third ◽  
Additive Error ◽  
N Input ◽  
Output Probability ◽  
Small Additive

We study the classical simulatability of commuting quantum circuits with n input qubits and O(log n) output qubits, where a quantum circuit is classically simulatable if its output probability distribution can be sampled up to an exponentially small additive error in classical polynomial time. Our main result is that there exists a commuting quantum circuit that is not classically simulatable unless the polynomial hierarchy collapses to the third level. This is the first formal evidence that a commuting quantum circuit is not classically simulatable even when the number of output qubits is O(log n). Then, we consider a generalized version of the circuit and clarify the condition under which it is classically simulatable. Lastly, using a proof similar to that of the main result, we provide an evidence that a slightly extended Clifford circuit is not classically simulatable.

scholarly journals Quantum circuits and Spin(3n) groups

2015 ◽  
Vol 15 (3&4) ◽  
pp. 235-259
Author(s):  
Alexander Yu. Vlasov
Keyword(s):  
Tensor Product ◽  
Unitary Transformation ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Spin Groups ◽  
Classical Simulation ◽  
The Matrix ◽  
Universal Quantum ◽  
Usual Quantum

All quantum gates with one and two qubits may be described by elements of Spin groups due to isomorphisms Spin(3)\isomSU(2) and Spin(6)\isomSU(4). However, the group of n-qubit gates SU(2^n) for n>2 has bigger dimension than Spin(3n). A quantum circuit with one- and two-qubit gates may be used for construction of arbitrary unitary transformation SU(2^n). Analogously, the `$Spin(3n)$ circuits' are introduced in this work as products of elements associated with one- and two-qubit gates with respect to the above-mentioned isomorphisms. The matrix tensor product implementation of the Spin(3n) group together with relevant models by usual quantum circuits with 2n qubits are investigated in such a framework. A certain resemblance with well-known sets of non-universal quantum gates (e.g., matchgates, noninteracting-fermion quantum circuits) related with Spin(2n) may be found in presented approach. Finally, a possibility of the classical simulation of such circuits in polynomial time is discussed.

scholarly journals Estimating Gibbs partition function with quantum Clifford sampling

2022 ◽  
Author(s):  
Yusen Wu ◽  
Jingbo B Wang
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Sampling Technique ◽  
Quantum Circuit ◽  
Quantum Systems ◽  
Quantum Circuits ◽  
Partition Functions ◽  
Stochastic Matrices ◽  
Many Body ◽  
Quantum Devices

Abstract The partition function is an essential quantity in statistical mechanics, and its accurate computation is a key component of any statistical analysis of quantum system and phenomenon. However, for interacting many-body quantum systems, its calculation generally involves summing over an exponential number of terms and can thus quickly grow to be intractable. Accurately and efficiently estimating the partition function of its corresponding system Hamiltonian then becomes the key in solving quantum many-body problems. In this paper we develop a hybrid quantumclassical algorithm to estimate the partition function, utilising a novel Clifford sampling technique. Note that previous works on quantum estimation of partition functions require O(1/ε√∆)-depth quantum circuits [17, 23], where ∆ is the minimum spectral gap of stochastic matrices and ε is the multiplicative error. Our algorithm requires only a shallow O(1)-depth quantum circuit, repeated O(n/ε2) times, to provide a comparable ε approximation. Shallow-depth quantum circuits are considered vitally important for currently available NISQ (Noisy Intermediate-Scale Quantum) devices.

scholarly journals Connectivity matrix model of quantum circuits and its application to distributed quantum circuit optimization

2021 ◽  
Vol 20 (7) ◽  
Author(s):  
Ismail Ghodsollahee ◽  
Zohreh Davarzani ◽  
Mariam Zomorodi ◽  
Paweł Pławiak ◽  
Monireh Houshmand ◽  
...  
Keyword(s):  
Quantum Computation ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Connectivity Matrix ◽  
Circuit Optimization ◽  
Novel Approach ◽  
The Matrix ◽  
Minimum Number ◽  
Two Phases

AbstractAs quantum computation grows, the number of qubits involved in a given quantum computer increases. But due to the physical limitations in the number of qubits of a single quantum device, the computation should be performed in a distributed system. In this paper, a new model of quantum computation based on the matrix representation of quantum circuits is proposed. Then, using this model, we propose a novel approach for reducing the number of teleportations in a distributed quantum circuit. The proposed method consists of two phases: the pre-processing phase and the optimization phase. In the pre-processing phase, it considers the bi-partitioning of quantum circuits by Non-Dominated Sorting Genetic Algorithm (NSGA-III) to minimize the number of global gates and to distribute the quantum circuit into two balanced parts with equal number of qubits and minimum number of global gates. In the optimization phase, two heuristics named Heuristic I and Heuristic II are proposed to optimize the number of teleportations according to the partitioning obtained from the pre-processing phase. Finally, the proposed approach is evaluated on many benchmark quantum circuits. The results of these evaluations show an average of 22.16% improvement in the teleportation cost of the proposed approach compared to the existing works in the literature.

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