A number of quantum algorithms have been performed on small quantum computers; these include Shor's prime factorization algorithm, error correction, Grover's search algorithm and a number of analog and digital quantum simulations. Because of the number of gates and qubits necessary, however, digital quantum particle simulations remain untested. A contributing factor to the system size required is the number of ancillary qubits needed to implement matrix exponentials of the potential operator. Here, we show that a set of tunneling problems may be investigated with no ancillary qubits and a cost of one single-qubit operator per time step for the potential evolution, eliminating at least half of the quantum gates required for the algorithm and more than that in the general case. Such simulations are within reach of current quantum computer architectures.
A number of quantum algorithms have been performed on small quantum computers; these include Shor's prime factorization algorithm, error correction, Grover's search algorithm and a number of analog and digital quantum simulations. Because of the number of gates and qubits necessary, however, digital quantum particle simulations remain untested. A contributing factor to the system size required is the number of ancillary qubits needed to implement matrix exponentials of the potential operator. Here, we show that a set of tunneling problems may be investigated with no ancillary qubits and a cost of one single-qubit operator per time step for the potential evolution, eliminating at least half of the quantum gates required for the algorithm and more than that in the general case. Such simulations are within reach of current quantum computer architectures.
It is well known that quantum computers are superior to classical computers in efficiently simulating quantum systems. Here we report the first experimental simulation of quantum tunneling through potential barriers, a widespread phenomenon of a unique quantum nature, via NMR techniques. Our experiment is based on a digital particle simulation algorithm and requires very few spin-1/2 nuclei without the need of ancillary qubits. The occurrence of quantum tunneling through a barrier, together with the oscillation of the state in potential wells, are clearly observed through the experimental results. This experiment has clearly demonstrated the possibility to observe and study profound physical phenomena within even the reach of small quantum computers.
Models of Grover’s search algorithm is reviewed to build the foundation for the other algorithms. Thereafter, some preliminary modifications of the original algorithms by others are stated, that increases the applicability of the search procedure. A general quantum computation on an isolated system can be represented by a unitary matrix. In order to execute such a computation on a quantum computer, it is common to decompose the unitary into a quantum circuit, i.e., a sequence of quantum gates that can be physically implemented on a given architecture. There are different universal gate sets for quantum computation. Here we choose the universal gate set consisting of CNOT and single-qubit gates. We measure the cost of a circuit by the number of CNOT gates as they are usually more difficult to implement than single qubit gates and since the number of single-qubit gates is bounded by about twice the number of CNOT’s.
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.
Classical simulations of quantum computers give us an insight into various things such as what quantum algorithms can achieve, whether it is possible to verify that they function as postulated, the difficulties that have to be overcome before quantum computers can be realized, as well as how we can handle these difficulties. We build a Python library to simulate a quantum computer that can perform all quantum algorithms by defining an universal gate set, from which all quantum gates – and thereby all circuits – can be constructed. We implement two algorithms that both highlight two important aspects of quantum computing: Grover’s quantum search algorithm, which demonstrates the efficiency of quantum algorithms and their superiority over their classical counterparts by searching an unsorted list quadratically faster; and an error correcting code, the Shor code, which highlights the cost of correcting the possible errors in a quantum computer.We test the rigidity of Grover’s algorithm by introducing errors without correction, and find that the algorithm shows resilience to smaller 1-qubit Pauli errors, but looses its efficiency under larger errors and thus the need for Pauli error correction arise.
We present an efficient algorithm for simulating open quantum systems dynamics described by the Lindblad master equation on quantum computers, addressing key challenges in the field. In contrast to existing approaches, our method achieves two significant advancements. First, we employ a repetition of unitary gates on a set of n system qubits and, remarkably, only a single ancillary bath qubit representing the environment. It follows that, for the typical case of m locality of the Lindblad operators, we reach an exponential improvement of the number of ancilla in terms of m and up to a polynomial improvement in ancilla overhead for large n with respect to other approaches. Although stochasticity is introduced, requiring multiple circuit realizations, the sampling overhead is independent of the system size. Second, we show that, under fixed accuracy conditions, our algorithm enables a reduction in the number of Trotter steps compared to other approaches, substantially decreasing circuit depth. These advancements hold particular significance for near-term quantum computers, where minimizing both width and depth is critical due to inherent noise in their dynamics. Published by the American Physical Society 2024
The factorization of large digit integers in polynomial time is a challenging computational task to decipher. The development of Shor’s algorithm sparked a new resolution for solving the factorization problem. However, putting Shor’s algorithm into use in real-world situations presents major difficulties. The algorithm largely depends on the availability of large-scale, fault-tolerant quantum computers, which are not available at present. The need for qubit coherence and error correction makes the algorithm susceptible to noise and decoherence, hindering its practical realization. Therefore, exploring alternative quantum factorization algorithms and investing in quantum computing hardware advancements are vital steps toward overcoming these drawbacks and harnessing the full potential of quantum computing for factorization tasks. This article explores an alternative method of converting the factorization problem into an optimization problem using appropriate analytic algebra. The generalized Grover’s protocol is used to increase the amplitude of the necessary states and, in turn, help in the execution of the quantum factorization of tetra and penta primes as a proof of concept for different integers, including 875, 1 269 636 549 803, and 4375, using three and four qubits of IBMQ Perth (a seven-qubit processor). The fidelity of the quantum factorization protocol with the IBMQ Perth qubits was near unity. A generalization of the method is provided at the end for implementing factorization problems in various cases.