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 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 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.

scholarly journals Error mitigation with Clifford quantum-circuit data

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 592
Author(s):  
Piotr Czarnik ◽  
Andrew Arrasmith ◽  
Patrick J. Coles ◽  
Lukasz Cincio
Keyword(s):  
State Energy ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Training Data ◽  
Quantum Circuits ◽  
Accurate Estimation ◽  
Error Mitigation ◽  
Order Of Magnitude ◽  
Quantum Observables ◽  
Near Term

Achieving near-term quantum advantage will require accurate estimation of quantum observables despite significant hardware noise. For this purpose, we propose a novel, scalable error-mitigation method that applies to gate-based quantum computers. The method generates training data {Xinoisy,Xiexact} via quantum circuits composed largely of Clifford gates, which can be efficiently simulated classically, where Xinoisy and Xiexact are noisy and noiseless observables respectively. Fitting a linear ansatz to this data then allows for the prediction of noise-free observables for arbitrary circuits. We analyze the performance of our method versus the number of qubits, circuit depth, and number of non-Clifford gates. We obtain an order-of-magnitude error reduction for a ground-state energy problem on 16 qubits in an IBMQ quantum computer and on a 64-qubit noisy simulator.

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.

Machine invention of quantum computing circuits by means of genetic programming

2008 ◽  
Vol 22 (3) ◽  
pp. 275-283 ◽  
Author(s):  
Lee Spector ◽  
Jon Klein
Keyword(s):  
Quantum Computing ◽  
Genetic Programming ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Practical Significance ◽  
Programming System ◽  
Computer Simulator ◽  
Classical Quantum ◽  
Better Than

AbstractWe demonstrate the use of genetic programming in the automatic invention of quantum computing circuits that solve problems of potential theoretical and practical significance. We outline a developmental genetic programming scheme for such applications; in this scheme the evolved programs, when executed, build quantum circuits and the resulting quantum circuits are then tested for “fitness” using a quantum computer simulator. Using the PushGP genetic programming system and the QGAME quantum computer simulator we demonstrate the invention of a new, better than classical quantum circuit for the two-oracle AND/OR problem.

scholarly journals Quick Quantum Circuit Simulation

2021 ◽  
Vol 3 (4) ◽  
Author(s):  
Daniel Evans
Keyword(s):  
Quantum State ◽  
Quantum Computer ◽  
Circuit Simulation ◽  
Matrix Multiplication ◽  
Quantum Circuit ◽  
Machine Language ◽  
Unitary Matrix ◽  
Initial State ◽  
Final State ◽  
Quantum Uncertainty

Quick Quantum Circuit Simulation (QQCS) is a software system for computing the result of a quantum circuit using a notation that derives directly from the circuit, expressed in a single input line. Quantum circuits begin with an initial quantum state of one or more qubits, which are the quantum analog to classical bits. The initial state is modified by a sequence of quantum gates, quantum machine language instructions, to get the final state. Measurements are made of the final state and displayed as a classical binary result. Measurements are postponed to the end of the circuit because a quantum state collapses when measured and produces probabilistic results, a consequence of quantum uncertainty. A circuit may be run many times on a quantum computer to refine the probabilistic result. Mathematically, quantum states are 2n -dimensional vectors over the complex number field, where n is the number of qubits. A gate is a 2n ×2n unitary matrix of complex values. Matrix multiplication models the application of a gate to a quantum state. QQCS is a mathematical rendering of each step of a quantum algorithm represented as a circuit, and as such, can present a trace of the quantum state of the circuit after each gate, compute gate equivalents for each circuit step, and perform measurements at any point in the circuit without state collapse. Output displays are in vector coefficients or Dirac bra-ket notation. It is an easy-to-use educational tool for students new to quantum computing.

scholarly journals Quantum advantage of unitary Clifford circuits with magic state inputs

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180427 ◽  
Author(s):  
Mithuna Yoganathan ◽  
Richard Jozsa ◽  
Sergii Strelchuk
Keyword(s):  
Polynomial Hierarchy ◽  
Efficient Computation ◽  
Computational Power ◽  
Average Case ◽  
Multiplicative Error ◽  
Additive Error ◽  
Universal Computation ◽  
Classical Simulation ◽  
Broad Variety ◽  
The Way

We study the computational power of unitary Clifford circuits with solely magic state inputs (CM circuits), supplemented by classical efficient computation. We show that CM circuits are hard to classically simulate up to multiplicative error (assuming polynomial hierarchy non-collapse), and also up to additive error under plausible average-case hardness conjectures. Unlike other such known classes, a broad variety of possible conjectures apply. Along the way, we give an extension of the Gottesman–Knill theorem that applies to universal computation, showing that for Clifford circuits with joint stabilizer and non-stabilizer inputs, the stabilizer part can be eliminated in favour of classical simulation, leaving a Clifford circuit on only the non-stabilizer part. Finally, we discuss implementational advantages of CM circuits.

scholarly journals Automated Quantum Hardware Selection for Quantum Workflows

Electronics ◽  
2021 ◽  
Vol 10 (8) ◽  
pp. 984
Author(s):  
Benjamin Weder ◽  
Johanna Barzen ◽  
Frank Leymann ◽  
Marie Salm
Keyword(s):  
Quantum Computer ◽  
Quantum Algorithm ◽  
Software Tool ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Quantum Computers ◽  
Workflow Model ◽  
Selection For ◽  
Selection Of

The execution of a quantum algorithm typically requires various classical pre- and post-processing tasks. Hence, workflows are a promising means to orchestrate these tasks, benefiting from their reliability, robustness, and features, such as transactional processing. However, the implementations of the tasks may be very heterogeneous and they depend on the quantum hardware used to execute the quantum circuits of the algorithm. Additionally, today’s quantum computers are still restricted, which limits the size of the quantum circuits that can be executed. As the circuit size often depends on the input data of the algorithm, the selection of quantum hardware to execute a quantum circuit must be done at workflow runtime. However, modeling all possible alternative tasks would clutter the workflow model and require its adaptation whenever a new quantum computer or software tool is released. To overcome this problem, we introduce an approach to automatically select suitable quantum hardware for the execution of quantum circuits in workflows. Furthermore, it enables the dynamic adaptation of the workflows, depending on the selection at runtime based on reusable workflow fragments. We validate our approach with a prototypical implementation and a case study demonstrating the hardware selection for Simon’s algorithm.

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 Classification with Quantum Neural Networks on Near Term Processors

2020 ◽  
Author(s):  
Edward Farhi ◽  
Hartmut Neven
Keyword(s):  
Quantum State ◽  
Quantum Computer ◽  
Binary Classification ◽  
Quantum Circuit ◽  
Input State ◽  
Data Sets ◽  
Quantum Neural Network ◽  
Small Quantum ◽  
Classical Simulation ◽  
Near Term

We introduce a quantum neural network, QNN, that can represent labeled data, classical or quantum, and be trained by supervised learning. The quantum circuit consists of a sequence of parameter dependent unitary transformations which acts on an input quantum state. For binary classification a single Pauli operator is measured on a designated readout qubit. The measured output is the quantum neural network’s predictor of the binary label of the input state. We show through classical simulation that parameters can be found that allow the QNN to learn to correctly distinguish the two data sets. We then discuss presenting the data as quantum superpositions of computational basis states corresponding to different label values. Here we show through simulation that learning is possible. We consider using our QNN to learn the label of a general quantum state. By example we show that this can be done. Our work is exploratory and relies on the classical simulation of small quantum systems. The QNN proposed here was designed with near-term quantum processors in mind. Therefore it will be possible to run this QNN on a near term gate model quantum computer where its power can be explored beyond what can be explored with simulation.

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