The approach we considered so far for computing using quantum mechanics relies on preparing a suitable initial state, applying single and two-qubit quantum gates, and performing measurements. It is known as the circuit model of quantum computing. Howeveer, this is not the only approach. In this chapter, we will cover a few alternatives. Some still rely on circuits to some extend, but they either explore alternative paradigms of fault tolerance, utilize protocols that do not require entangling gates, or avoid multi-shot computations. One of these alternative approaches is entirely based on measurements. Another requires fine tunning gates and repeating measurments.
The approach we considered so far for computing using quantum mechanics relies on preparing a suitable initial state, applying single and two-qubit quantum gates, and performing measurements. It is known as the circuit model of quantum computing. Howeveer, this is not the only approach. In this chapter, we will cover a few alternatives. Some still rely on circuits to some extend, but they either explore alternative paradigms of fault tolerance, utilize protocols that do not require entangling gates, or avoid multi-shot computations. One of these alternative approaches is entirely based on measurements. Another requires fine tunning gates and repeating measurments.
Introduction In the standard model of quantum computation [D95], some simple quantum state is prepared and evolved by a partially ordered set of unitary gates, and finally is measured in the computation basis. Up until the late 1990s, transformations that deviate from unitarity were commonly associated with irreversibility, and thus incoherence and harm induced in the quantum data of interest. However, this viewpoint was challenged by three notable results: (1) teleportation [BBC+93], (2) syndrome measurements [S95, S96e], and (3) the fault-tolerant Toffoli and π/8-gate constructions [S96b, BMP+99]. In these schemes, well-designed measurements project the quantum data of interest onto a subspace of the original ambient space for each outcome, up to some known unitary transformation that is either reversible (and harmless) or intended. Subsequently, researchers found many useful error-correction procedures and unitary gates that rely heavily on measurements. Two “measurement models” of quantum computation were proposed in the early 2000s that make extreme use of measurements. In the cluster model [RB01], any quantum computation can be performed by applying a partially time-ordered set of single-qubit measurements on a certain entangled state. The state must be of appropriate size, but is otherwise independent of the computation. The related graph state model [RBB03] uses an initial state that depends on the computation itself. In the seemingly different teleportation-based model [N03, L01a, L04], any quantum computation can be implemented by applying one- or two-qubit nondemolition measurements on an arbitrary initial state, so as to “teleport” each gate in the simulated circuit.
We discuss the building blocks of a universal quantum computer within the circuit model of computation and how this is implemented using superconducting quantum circuits. In particular, we discuss, one by one, the creation of quantum registers, resetting of quantum bits, qubit measurements, single-qubit operations, and universal two-qubit gates, and how these are all implemented using the tools from earlier chapters. We discuss how to calibrate the errors in the qubits and in the operations, assigning them complete descriptions via positive maps. We explain how these errors can be corrected and how to implement a fault-tolerant quantum computer, focusing on the paradigm of stabilizer codes and the surface code in particular. We close with a discussion on the outlook for quantum computers in the near term and the NISQ paradigm of computation.
Quantum computing is an emerging technology that has the potential to change the perspectives and applications of computing in general. A wide range of applications are enabled: from faster algorithmic solutions of classically still difficult problems to theoretically more secure communication protocols. A quantum computer uses the quantum mechanical effects of particles or particle-like systems, and a major similarity between quantum and classical computers consists of both being abstracted as information processing machines. Whereas a classical computer operates on classical digital information, the quantum computer processes quantum information, which shares similarities with analog signals. One of the central differences between the two types of information is that classical information is more fault-tolerant when compared to its quantum counterpart. Faults are the result of the quantum systems being interfered by external noise, but during the last decades quantum error correction codes (QECC) were proposed as methods to reduce the effect of noise. Reliable quantum circuits are the result of designing circuits that operate directly on encoded quantum information, but the circuit’s reliability is also increased by supplemental redundancies, such as sub-circuit repetitions. Reliable quantum circuits have not been widely used, and one of the major obstacles is their vast associated resource overhead, but recent quantum computing architectures show promising scalabilities. Consequently the number of particles used for computing can be more easily increased, and that the classical control hardware (inherent for quantum computation) is also more reliable. Reliable quantum circuits haev been investigated for almost as long as general quantum computing, but their limited adoption (until recently) has not generated enough interest into their systematic design. The continuously increasing practical relevance of reliability motivates the present thesis to investigate some of the first answers to questions related to the background and the methods forming a reliable quantum circuit design stack. The specifics of quantum circuits are analysed from two perspectives: their probabilistic behaviour and their topological properties when a particular class of QECCs are used. The quantum phenomena, such as entanglement and superposition, are the computational resources used for designing quantum circuits. The discrete nature of classical information is missing for quantum information. An arbitrary quantum system can be in an infinite number of states, which are linear combinations of an exponential number of basis states. Any nontrivial linear combination of more than one basis states is called a state superposition. The effect of superpositions becomes evident when the state of the system is inferred (measured), as measurements are probabilistic with respect to their output: a nontrivial state superposition will collapse to one of the component basis states, and the measurement result is known exactly only after the measurement. A quantum system is, in general, composed from identical subsystems, meaning that a quantum computer (the complete system) operates on multiple similar particles (subsystems). Entanglement expresses the impossibility of separating the state of the subsystems from the state of the complete system: the nontrivial interactions between the subsystems result into a single indivisible state. Entanglement is an additional source of probabilistic behaviour: by measuring the state of a subsystem, the states of the unmeasured subsystems will probabilistically collapse to states from a well defined set of possible states. Superposition and entanglement are the building blocks of quantum information teleportation protocols, which in turn are used in state-of-the-art fault-tolerant quantum computing architectures. Information teleportation implies that the state of a subsystem is moved to a second subsystem without copying any information during the process. The probabilistic approach towards the design of quantum circuits is initiated by the extension of classical test and diagnosis methods. Quantum circuits are modelled similarly to classical circuits by defining gate-lists, and missing quantum gates are modelled by the single missing gate fault. The probabilistic approaches towards quantum circuits are facilitated by comparing these to stochastic circuits, which are a particular type of classical digital circuits. Stochastic circuits can be considered an emulation of analogue computing using digital components. A first proposed design method, based on the direct comparison, is the simulation of quantum circuits using stochastic circuits by mapping each quantum gate to a stochastic computing sub-circuit. The resulting stochastic circuit is compiled and simulated on FPGAs. The obtained results are encouraging and illustrate the capabilities of the proposed simulation technique. However, the exponential number of possible quantum basis states was translated into an exponential number of stochastic computing elements. A second contribution of the thesis is the proposal of test and diagnosis methods for both stochastic and quantum circuits. Existing verification (tomographic) methods of quantum circuits were targeting the reconstruction of the gate-lists. The repeated execution of the quantum circuit was followed by different but specific measurement at the circuit outputs. The similarities between stochastic and quantum circuits motivated the proposal of test and diagnosis methods that use a restricted set of measurement types, which minimise the number of circuit executions. The obtained simulation results show that the proposed validation methods improve the feasibility of quantum circuit tomography for small and medium size circuits. A third contribution of the thesis is the algorithmic formalisation of a problem encountered in teleportation-based quantum computing architectures. The teleportation results are probabilistic and require corrections represented as quantum gates from a particular set. However, there are known commutation properties of these gates with the gates used in the circuit. The corrections are not applied as dynamic gate insertions (during the circuit’s execution) into the gate-lists, but their effect is tracked through the circuit, and the corrections are applied only at circuit outputs. The simulation results show that the algorithmic solution is applicable for very large quantum circuits. Topological quantum computing (TQC) is based on a class of fault-tolerant quantum circuits that use the surface code as the underlying QECC. Quantum information is encoded in lattice-like structures and error protection is enabled by the topological properties of the lattice. The 3D structure of the lattice allows TQC computations to be visualised similarly to knot diagrams. Logical information is abstracted as strands and strand interactions (braids) represent logical quantum gates. Therefore, TQC circuits are abstracted using a geometrical description, which allows circuit input-output transformations (correlations) to be represented as geometric sub-structures. TQC design methods were not investigated prior to this work, and the thesis introduces the topological computational model by first analysing the necessary concepts. The proposed TQC design stack follows a top-down approach: an arbitrary quantum circuit is decomposed into the TQC supported gate set; the resulting circuit is mapped to a lattice of appropriate dimensions; relevant resulting topological properties are extracted and expressed using graphs and Boolean formulas. Both circuit representations are novel and applicable to TQC circuit synthesis and validation. Moreover, the Boolean formalism is broadened into a formal mechanism for proving circuit correctness. The thesis introduces TQC circuit synthesis, which is based on a novel logical gate geometric description, whose formal correctness is demonstrated. Two synthesis methods are designed, and both use a general planar representation of the circuit. Initial simulation results demonstrate the practicality and performance of the methods. An additional group of proposed design methods solves the problem of automatic correlation construction. The methods use validity criteria which were introduced and analysed beforehand in the thesis. Input-output correlations existing in the circuit are inferred using both the graph and the Boolean representation. The thesis extends the TQC state-of-the-art by recognising the importance of correlations in the validation process: correlation construction is used as a sub-routine for TQC circuit validation. The presented cross-layer validation procedure is useful when investigating both the QECC and the circuit, while a second proposed method is QECC-independent. Both methods are scalable and applicable even to very large circuits. The thesis completes with the analysis of TQC circuit identities, where the developed Boolean formalism is used. The proofs of former known circuit identities were either missing or complex, and the presented approach reduces the length of the proofs and represents a first step towards standardising them. A new identity is developed and detailed during the process of illustrating the known circuit identities. Reliable quantum circuits are a necessity for quantum computing to become reality, and specialised design methods are required to support the quest for scalable quantum computers. This thesis used a twofold approach towards this target: firstly by focusing on the probabilistic behaviour of quantum circuits, and secondly by considering the requirements of a promising quantum computing architecture, namely TQC. Both approaches resulted in a set of design methods enabling the investigation of reliable quantum circuits. The thesis contributes with the proposal of a new quantum simulation technique, no
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.
Fault tolerant quantum computers repetitively apply a four-step procedure: First, perform a few one and two-qubit quantum gates. Second, perform a syndrome measurement on a subset of the qubits. Third, perform fast classical computations to establish if and where errors occurred. And, fourth, correct the errors with a correction step. The next iteration applies the same procedure with new one and two-qubit gates. Even though current error-rates prohibit this procedure to work and fault tolerant quantum computing remains a distant goal, the same procedure can already prove useful today. In this work we make use of this four-step scheme not to carry out fault-tolerant computations, but to enhance short, constant -depth, quantum circuits that perform 1 qubit gates and nearest−neighbor 2 qubit gates.We introduce a new computational model called Local Alternating Quantum Classical Computations (LAQCC) . In this model, qubits are placed in a grid and they can only interact with their direct neighbors; the quantum circuits are of constant depth with intermediate measurements; a classical controller can perform log-depth computations on these intermediate measurement outcomes and control future quantum operations based on the outcome. This model fits naturally between quantum algorithms in the NISQ era and full-fledged fault-tolerant quantum computation. We first prove that any Clifford circuit has an equivalent LAQCC circuit, and that any LAQCC circuit can be simulated by a QNC1 circuit. Next, we conjecture the non-simulatability of LAQCC by showing that LAQCC contains the class of Instantaneous Quantum Polynomial-time circuits. We also show that any LAQCC circuit with polynomial-sized quantum circuits and unbounded classical computations is contained in the class of quantum circuits equipped with post-selection gates with respect to the task of state preparation. We continue by presenting LAQCC implementations of different subroutines, including OR-gates, quantum Fourier transforms and Threshold gates. These subroutines prove vital in constructing three state preparation routines in the main part of this work. Preparing a uniform superposition uses constant-depth arithmetic gates, combined with an exact Grover implementation by Long. For the W -state, we employ a compress-uncompress method to switch between a binary and one-hot encoding. This method extends to the more generalized Dicke-states, the superposition of n -bit strings of Hamming weight k , for k=O(n) , but fails for higher k due to the birthday paradox. We extend this protocol to a protocol that prepares many-body scar states, highly excited states with low entanglement and longer coherence times than states with the same energy density. We present a circuit for preparing Dicke-states for larger k requiring log-depth circuits that maps between the factoradic number system and the combinatorial number system.
Fault-tolerant modular quantum computing requires stabilizer measurements across the modules in a quantum network. For this, entangled states of high quality and rate must be distributed. Currently, two main types of entanglement distribution protocols exist, namely emission-based and scattering-based, each with its own advantages and drawbacks. On the one hand, scattering-based protocols with cavities or waveguides are fast but demand stringent hardware such as high-efficiency integrated circulators or strong waveguide coupling. On the other hand, emission-based platforms are experimentally feasible but so far rely on Bell-pair fusion with extensive use of slow two-qubit memory gates, limiting thresholds to ≈0.16%. Here, we consider a fully distributed surface code using emission-based entanglement schemes that generate GHZ states in a single shot, i.e., without the need for Bell-pair fusions. We show that our optical setup produces Bell pairs, W states, and GHZ states, enabling both memory-based and optical protocols for distilling high-fidelity GHZ states with significantly improved success rates. Furthermore, we introduce protocols that completely eliminate the need for memory-based two-qubit gates, achieving thresholds of ≈0.19% with modest hardware enhancements, increasing to above ≈0.24% with photon-number-resolving detectors. These results show the feasibility of emission-based architectures for scalable fault-tolerant operation.