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Quantum circuits

Riccardo Manenti, Mário Motta

Published Aug 25, 2023• 0 citations
Publisher

Abstract

Abstract In this chapter, we describe the circuit model of quantum computation, focusing on the initialization of a set of qubits, their unitary evolution, and final projective measurement. We outline Di Vincenzo’s criteria for constructing a functional quantum computer. We describe single-qubit gates and measurements, and we present two-qubit gates. We prove that single-qubit gates and a particular two-qubit gate, the CNOT gate, are universal for quantum computation. In the last part of the chapter, we present the Clifford group of unitary transformations that stabilize the Pauli group, and the Gottesman-Knill theorem.

DOI
10.1093/oso/9780198787488.003.0005

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Abstract In this chapter, we describe the circuit model of quantum computation, focusing on the initialization of a set of qubits, their unitary evolution, and final projective measurement. We outline Di Vincenzo’s criteria for constructing a functional quantum computer. We describe single-qubit gates and measurements, and we present two-qubit gates. We prove that single-qubit gates and a particular two-qubit gate, the CNOT gate, are universal for quantum computation. In the last part of the chapter, we present the Clifford group of unitary transformations that stabilize the Pauli group, and the Gottesman-Knill theorem.
We are still at the very beginnings of physical implementations of quantum computers. The devices listed below have been used successfully to entangle two qubits (at most!), except for NMR which has gone up to 7 qubits. It is still premature to try to predict which device will prove most effective for building a quantum computer capable of dealing with several hundred qubits (if such a computer someday exists); perhaps it will be something new, not on this list at all. In any case, it would be as foolish to predict that such a computer will not be available by 2050 as to predict the contrary. The storage and processing of quantum information requires physical systems possessing the following properties (di Vincenzo criteria): (i) they must be scalable, that is, capable of being extended to a sufficient number of qubits, with well defined qubits; (ii) they must have qubits which can be initialized in the state |0〉; (iii) they must have qubits which are carried by physical states of sufficiently long lifetime, so as to ensure that the quantum states remain coherent throughout the calculation; (iv) they must possess a set of universal quantum gates: unitary transformations on individual qubits and a cNOT gate, which are obtained by controlled manipulations; (v) there must be an efficient procedure for measuring the state of the qubits at the end of the calculation ( readout of the results).
With the anticipated end of Moore’s law for integrated circuits [3, 4] fast approaching and continued advances in low-power electronics, interest in quantum computing has increased. This shifts the focus from deterministic logical circuits to potentially more powerful circuits based on controllable quantum systems. In this column, we present a mathematical tour of the quantum circuit model, beginning with reversible logic circuits and expanding to quantum circuits, gates, and measurement. We highlight quantum mechanical phenomena such as superposition, entanglement, and measurement, review the Gottesman-Knill theorem, which states that some subclasses of quantum operations can be simulated eciently on a classical computer, and describe sets of quantum gates that are universal for quantum computation.
Abstract The standard stabilizer formalism provides a setting to show that quantum computations restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman–Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of SU(d) transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results provide better insights into the structure of Gottesman–Knill-type results, consequently allowing us to draw a sharper line between quantum and classical computation.
Abstract In this chapter, we shift our perspective from seeing the unitary evolution of a quantum state as a physical process that ‘just happens’ to instead seeing it as an executable operation in a programmable computer. To do this, we introduce the quantum circuit (or gate) model, which subsequently allows us to present and analyse quantum algorithms and quantum communication protocols. The universality of quantum computing – that every quantum mechanical system (composed of qubits) can be represented as such – is established, as is the closely related notion of universal gatesets. The chapter also briefly introduces some important topics in quantum compilation, in which some quantum circuit is rewritten as a functionally equivalent alternative that is more suitable (in some sense) for running on a particular quantum computer.
Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G tensor G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups G_n generated by X and the n^th root of Z.
In Section 1.3, we introduced the circuit model of (classical) computation. We restricted attention to reversible circuits since they can simulate any non-reversible circuit with modest overhead. This model can be generalized to a model of quantum circuits. In the quantum circuit model, we have logical qubits carried along ‘wires’, and quantum gates that act on the qubits. A quantum gate acting on n qubits has the input qubits carried to it by n wires, and n other wires carry the output qubits away from the gate. A quantum circuit is often illustrated schematically by a circuit diagram as shown in Figure 4.1. The wires are shown as horizontal lines, and we imagine the qubits propagating along the wires from left to right in time. The gates are shown as rectangular blocks. For convenience, we will restrict attention to unitary quantum gates (which are also reversible). Recall from Section 3.5.3 that non-unitary (non-reversible) quantum operations can be simulated by unitary (reversible) quantum gates if we allow the possibility of adding an ancilla and of discarding some output qubits. A circuit diagram describing a superoperator being implemented using a unitary operator is illustrated in Figure 4.2. In the example of Figure 4.1, the 4-qubit state |ψi⟩= |0⟩⊗ |0⟩⊗ |0⟩⊗ |0⟩ enters the circuit at the left (recall we often write this state as |ψi⟩ = |0⟩|0⟩|0⟩|0⟩ or |ψi⟩ = |0000⟩.) These qubits are processed by the gates U1, U2, U3, and U4. At the output of the circuit we have the collective (possibly entangled) 4-qubit state |ψf⟩. A measurement is then made of the resulting state. The measurement will often be a simple qubit-by-qubit measurement in the computational basis, but in some cases may be a more general measurement of the joint state. A measurement of a single qubit in the computational basis is denoted on a circuit diagram by a small triangle, as shown in Figure 4.1 (there are other symbols used in the literature, but we adopt this one). The triangle symbol will be modified for cases in which there is a need to indicate different types of measurements.R50