We propose an efficient method for Hamiltonian simulation of multi-qubit quantum systems with special types of interaction. In our approach, the Hamiltonian of a
n-qubit system should be represented as a linear combination of the standard Pauli basis operators, and then decomposed into a sum of partial Hamiltonians, which are, in general, not Pauli operators and satisfy some anticommutation relations. For three types of Hamiltonians, which are invariant with respect to permutations of qubits, the effectiveness of the main algorithm in the three-qubit cluster model is shown by calculating the operator exponentials for these Hamiltonians in an explicit analytical form. We also calculate the density operator, partition function, entropy, and free energy of the cluster weakly coupled to a thermal environment. In our model, the cluster is in the Gibbs state in the temperature interval
0.1−2K, which corresponds to the operating range of modern quantum processors. It follows from our analysis that the thermodynamic properties of such systems strongly depend on the type of internal interaction of qubits in the cluster.
Unitary operations are the building blocks of quantum programs. Our task is to design efficient or optimal implementations of these unitary operations by employing the intrinsic physical resources of a given
n-qubit system. The most common versions of this task are known as Hamiltonian simulation and gate simulation, where Hamiltonian simulation can be seen as an infinitesimal version of the general task of gate simulation. We present a Lie-theoretic approach to Hamiltonian simulation and gate simulation. From this, we derive lower bounds on the time complexity in the
n-qubit case, generalizing known results to both even and odd
n. To achieve this we develop a generalization of the so-called magic basis for two-qubits. As a corollary, we note a connection to entanglement measures of concurrence-type.
These notes begin in Chapter 1 with a review of linear algebra and the postulates of quantum mechanics, leading to an explanation of single- and multi-qubit gates. Chapter 2 explores the challenge of constructing arbitrary quantum states from given initial states, and introduces circuits for building oracles. Chapter 3 presents foundational algorithms such as entanglement creation, quantum teleportation, Deutsch-Jozsa, Bernstein-Vazirani, and Simon's algorithm. Chapters 4 and 5 cover algorithms based on the quantum Fourier transform, including phase estimation, period finding, factoring, and logarithm computation. These chapters also include complexity analysis and detailed quantum circuits suitable for implementation in code. Chapter 6 introduces Grover's algorithm for quantum search and amplitude amplification, including its realization via Hamiltonian simulation and a method for derandomization. Chapter 7 discusses basic techniques for Hamiltonian simulation, such as Lie-Trotter decomposition, sparse Hamiltonians, and the linear combination of unitaries. It also provides example circuits for simulating Hamiltonians expressed as linear combinations of Pauli operators. Chapter 8 introduces variational quantum algorithms, and Chapter 9 presents an algorithm for simulating fermionic many-particle systems, with an emphasis on molecular Hamiltonians. It also outlines the key transformations needed to map a molecular Hamiltonian to a form suitable for simulation on a quantum computer.