A Quantum Computer is a new type of computer which can solve problems such as factoring and database search very efficiently. The usefulness of a quantum computer is limited by the effect of two different types of errors, decoherence and inaccuracies. In this paper we show the results of simulations of a quantum computer which consider both decoherence and inaccuracies. We simulate circuits which factor the numbers 15, 21, 35, and 57 as well as circuits which use database search to solve the circuit satisfaction problem. Our simulations show that the error rate per gate is on the order of 10^-6 for a trapped ion quantum computer whose noise is kept below pi/4096 per gate and with a decoherence rate of 10^-6. This is an important bound because previous studies have shown that a quantum computer can factor more efficiently than a classical computer if the error rate is of order 10^-6.
A Quantum Computer is a new type of computer which can solve problems such as factoring and database search very efficiently. The usefulness of a quantum computer is limited by the effect of two different types of errors, decoherence and inaccuracies. In this paper we show the results of simulations of a quantum computer which consider both decoherence and inaccuracies. We simulate circuits which factor the numbers 15, 21, 35, and 57 as well as circuits which use database search to solve the circuit satisfaction problem. Our simulations show that the error rate per gate is on the order of 10^-6 for a trapped ion quantum computer whose noise is kept below pi/4096 per gate and with a decoherence rate of 10^-6. This is an important bound because previous studies have shown that a quantum computer can factor more efficiently than a classical computer if the error rate is of order 10^-6.
Quantum computation is a new technology with the potential for providing exponential parallelism. The feasibility of a quantum computer is limited by two different types of errors, decoherence and inaccuracies. This dissertation uses simulation to investigate the feasibility of implementing one of the proposed realizations of a quantum computer, the trapped ion quantum computer. Because of the exponential complexity of a quantum computer, simulating one requires vast processing and memory resources. For this reason I develop several techniques to reduce the complexity of simulating a quantum computer. A parallel simulator is also used. Using the parallel simulator I am able to simulate circuits with close to 100,000 basic operations. The simulation studies show that the effectiveness of a quantum computer, without a mechanism for controlling errors, is severely limited. However the power of a quantum computer can be increased by using several techniques to mitigate the effects of error. Therefore a realizable quantum computer has three fundamental requirements: First, both hardware and software mechanisms must be used to correct errors as they occur. Second, the architecture of a quantum computer is very important; it must be designed to mitigate errors and it must be able to execute multiple operations in parallel. Last, quantum circuits must be constructed in an efficient manner, to minimize the total number of operations.
Recently Quantum Computation has generated a lot of interest due to the discovery of a quantum algorithm which can factor large numbers in polynomial time. The usefulness of a quantum com puter is limited by the effect of errors. Simulation is a useful tool for determining the feasibility of quantum computers in the presence of errors. The size of a quantum computer that can be simulat ed is small because faithfully modeling a quantum computer requires an exponential amount of storage and number of operations. In this paper we define simulation models to study the feasibility of quantum computers. The most detailed of these models is based directly on a proposed imple mentation. We also define less detailed models which are exponentially less complex but still pro duce accurate results. Finally we show that the two different types of errors, decoherence and inaccuracies, are uncorrelated. This decreases the number of simulations which must be per formed.
With current technologies, it seems to be very difficult to implement quantum computers with many qubits. It is therefore of importance to simulate quantum algorithms and circuits on the existing computers. However, for a large-size problem, the simulation often requires more computational power than is available from sequential processing. Therefore, simulation methods for parallel processors are required. We have developed a general-purpose simulator for quantum algorithms/circuits on the parallel computer (Sun Enterprise4500). It can simulate algorithms/circuits with up to 30 qubits. In order to test efficiency of our proposed methods, we have simulated Shor's factorization algorithm and Grover's database search, and we have analyzed robustness of the corresponding quantum circuits in the presence of both decoherence and operational errors. The corresponding results, statistics, and analyses are presented in this paper.
Abstract When quantum computing becomes a wide-spread commercial reality, Quantum Search Algorithms (QSA) and especially Grover’s QSA will inevitably be one of their main applications, constituting their cornerstone. Most of the literature assumes that the quantum circuits are free from decoherence. Practically, decoherence will remain unavoidable as is the Gaussian noise of classic circuits imposed by the Brownian motion of electrons, hence it may have to be mitigated. In this contribution, we investigate the effect of quantum noise on the performance of QSAs, in terms of their success probability as a function of the database size to be searched, when decoherence is modelled by depolarizing channels’ deleterious effects imposed on the quantum gates. Moreover, we employ quantum error correction codes for limiting the effects of quantum noise and for correcting quantum flips. More specifically, we demonstrate that, when we search for a single solution in a database having 4096 entries using Grover’s QSA at an aggressive depolarizing probability of 10−3, the success probability of the search is 0.22 when no quantum coding is used, which is improved to 0.96 when Steane’s quantum error correction code is employed. Finally, apart from Steane’s code, the employment of Quantum Bose-Chaudhuri-Hocquenghem (QBCH) codes is also considered.
Quantum technologies have shown immeasurable potential to effectively solve several information processing tasks such as prime number factorization, unstructured database search or complex macromolecule simulation. As a result of such capability to solve certain problems that are not classically tractable, quantum machines have the potential revolutionize the modern world via applications such as drug design, process optimization, unbreakable communications or machine learning. However, quantum information is prone to suffer from errors caused by the so-called decoherence, which describes the loss in coherence of quantum states associated to their interactions with the surrounding environment. This decoherence phenomenon is present in every quantum information task, be it transmission, processing or even storage of quantum information. Consequently, the protection of quantum information via quantum error correction codes (QECC) is of paramount importance to construct fully operational quantum computers. Understanding environmental decoherence processes and the way they are modeled is fundamental in order to construct effective error correction methods capable of protecting quantum information. In this thesis, the nature of decoherence is studied and mathematically modelled; and QECCs are designed and optimized so that they exhibit better error correction capabilities.
An ultimate goal of quantum computing is to perform calculations beyond the reach of any classical computer.It is therefore imperative that useful quantum computers be very difficult to simulate classically, otherwise classical computers could be used for the applications envisioned for the quantum ones.Perfect quantum computers are unarguably exponentially difficult to simulate: the classical resources required grow exponentially with the number of qubits N or the depth D of the circuit.This difficulty has triggered recent experiments on deep, random circuits that aim to demonstrate that quantum devices may already perform tasks beyond the reach of classical computing.These real quantum computing devices, however, suffer from many sources of decoherence and imprecision which limit the degree of entanglement that can actually be reached to a fraction of its theoretical maximum.They are characterized by an exponentially decaying fidelity F ∼ ð1 -ϵÞ ND with an error rate ϵ per operation as small as ≈1% for current devices with several dozen qubits or even smaller for smaller devices.In this work, we provide new insight on the computing capabilities of real quantum computers by demonstrating that they can be simulated at a tiny fraction of the cost that would be needed for a perfect quantum computer.Our algorithms compress the representations of quantum wave functions using matrix product states, which are able to capture states with low to moderate entanglement very accurately.This compression introduces a finite error rate ϵ so that the algorithms closely mimic the behavior of real quantum computing devices.The computing time of our algorithm increases only linearly with N and D in sharp contrast with exact simulation algorithms.We illustrate our algorithms with simulations of random circuits for qubits connected in both one-and twodimensional lattices.We find that ϵ can be decreased at a polynomial cost in computing power down to a minimum error ϵ ∞ .Getting below ϵ ∞ requires computing resources that increase exponentially with ϵ ∞ =ϵ.For a two-dimensional array of N ¼ 54 qubits and a circuit with control-Z gates, error rates better than state-of-the-art devices can be obtained on a laptop in a few hours.For more complex gates such as a SWAP gate followed by a controlled rotation, the error rate increases by a factor 3 for similar computing time.Our results suggest that, despite the high fidelity reached by quantum devices, only a tiny fraction ð∼10 -8 Þ of the system Hilbert space is actually being exploited.