It was recently shown (e-print quant-ph/9909074) that parasitic random interactions between the qubits in a quantum computer can induce quantum chaos and put into question the operability of a quantum computer. In this work I investigate whether already the interactions between the qubits introduced with the intention to operate the quantum computer may lead to quantum chaos. The analysis focuses on two well-known quantum algorithms, namely, Grover's search algorithm and the quantum Fourier transform. I show that in both cases the same very unusual combination of signatures from chaotic and from integrable dynamics arises.
It was recently shown (e-print quant-ph/9909074) that parasitic random interactions between the qubits in a quantum computer can induce quantum chaos and put into question the operability of a quantum computer. In this work I investigate whether already the interactions between the qubits introduced with the intention to operate the quantum computer may lead to quantum chaos. The analysis focuses on two well-known quantum algorithms, namely, Grover's search algorithm and the quantum Fourier transform. I show that in both cases the same very unusual combination of signatures from chaotic and from integrable dynamics arises.
How classical chaos emerges from the underlying quantum world is a fundamental problem in physics. The origin of this question is in the correspondence principle. Classical chaos arises due to non-linear dynamics, whereas quantum mechanics, driven by unitary evolution, is linear. The question that still remains is - what are the footprints of classical chaos in the quantum world? One can understand the quantum signatures of classical chaos by studying a quantum system whose classical analogue is chaotic. In this thesis, we use the quantum kicked top model of few qubits in the deep quantum regime to investigate signatures that can be considered as a precursor to chaos in the classical limit. In particular, we study out-of-time-ordered correlators (OTOCs) and Loschmidt echo, the two well-known dynamical diagnostics of chaos. We find vestiges of classical chaos even in such a deep quantum regime. Another arena where one can study the effects of chaos and randomness is quantum state tomography. We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators generated by a unitary evolution of an initial observable. The rate of information gain and reconstruction fidelity shows vestiges of chaos. As another contribution of this thesis, we have harnessed the power of randomness inherent in the maximally mixed state to give an efficient quantum algorithm to measure OTOCs. The protocol achieves an exponential speedup over the best known classical algorithm, provided the OTOC operator to be estimated admits an efficient gate decomposition. This protocol also helps benchmark unitary gates, which is important from the quantum computation and control perspective.
We consider the stability of a general quantum algorithm (QA) with respect to a fixed but unknown residual interaction between qubits, and show a surprising fact, namely that the average fidelity of quantum computation increases on decreasing the average time correlation function of the perturbing operator in sequences of consecutive quantum gates. Our thinking is applied to the quantum Fourier transformation, where an alternative `less regular' QA is devised, which is qualitatively more robust against static random residual n -qubit interaction.
We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos maps. For the Grover algorithm our numerical and analytical results show existence of regular and chaotic phases depending on the static imperfection strength ε. The critical border εc between two phases drops polynomially with the number of qubits nq as εc∼nq−3/2. In the regular phase (ε<εc) the algorithm remains robust against imperfections showing the efficiency gain εc/ε for ε>2−nq/2. In the chaotic phase (ε>εc) the algorithm is completely destroyed. The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix Theory (RMT). We also discuss a new gyroscopic quantum error correction method which allows to reduce the effect of static imperfections. In spite of this decay GYQEC allows to obtain a significant gain in the accuracy of quantum computations.
Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in n_q. In particular, quantum chaotic maps are efficient generators of multipartite entanglement among the qubits, close to that expected for random states. This review discusses several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer; (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics; (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment.
We show that in an isolated model of quantum computer the residual short-range inter-qubit couplings and fluctuations in individual qubit energies can lead to quantum chaos and quantum ergodicity of eigenstates. The onset of chaos results in the interaction induced dynamical thermalization and the occupation numbers well described by the Fermi–Dirac distribution. This thermalization destroys the noninteracting qubit structure and sets serious requirements for the quantum computer operability. We also show that a quantum computer operating with a small number of qubits can simulate the dynamical localization of classical chaos in a system described by the quantum sawtooth map model. The dynamics of the system is computed efficiently up to a time t ≥ l , and then the localization length l can be obtained with accuracy ν by means of order 1/ν 2 computer runs, followed by coarse grained projective measurements on the computational basis.
This paper investigates the intersection of quantum chaos and quantum computing control. It explores the potential influence of chaotic behavior within quantum systems on computational processes and proposes control strategies to mitigate these effects. The core argument centers on the recognition of quantum chaos as a crucial, yet often overlooked, element in the precise control required for reliable quantum computation. Through the application of chaos theory, this work analyzes the manifestation of chaotic dynamics in quantum systems, detailing the implications for qubit stability and gate fidelity. Specific control techniques, including feedback mechanisms and tailored Hamiltonian designs, are examined and discussed, ultimately aiming to harness, rather than suppress, chaotic behavior to potentially enhance certain quantum algorithms. The research contributes to a deeper understanding of the complex relationship between quantum systems and chaos, providing a foundation for the development of more robust and adaptable quantum control protocols. The study utilizes mathematical formalism to rigorously define and analyze the concepts explored, focusing on key equations governing quantum dynamics and chaotic systems.