Stabilizer Entanglement Distillation and Efficient Fault-Tolerant Encoders

Quantum computing is a multidisciplinary field at the intersectionof quantum physics, computer science, and mathematics, aiming to har-ness quantum mechanical phenomena for computational advantage. Thisarticle provides a comprehensive review of both the foundational princi-ples and advanced theoretical models of quantum computing. We beginwith the basic framework of quantum computation, introducing the con-cept of qubits as two-level quantum systems, the linear algebra of Hilbertspaces, Dirac’s bra–ket notation, and the postulates of quantum mechan-ics (state superposition, unitary evolution, and projective measurement).We then discuss quantum logic gates and circuits, highlighting how clas-sical reversible computation principles extend to universal quantum gatesets. Next, we survey key quantum algorithms (from Shor’s factoring toGrover’s searching and beyond) and the complexity-theoretic implicationsof quantum computers, contrasting the class BQP with classical complex-ity classes. We cover the theoretical foundations of quantum error cor-rection and fault tolerance, explaining how quantum information can beprotected from decoherence using redundancy and how fault-tolerant pro-tocols can, in principle, allow scalable quantum computation despite noisyhardware. We then explore advanced models and paradigms of quantumcomputation, including measurement-based quantum computing (one-wayquantum computing), topological quantum computation with anyons, adi-abatic quantum computing and quantum annealing, continuous-variablequantum computing, and related approaches. We also provide an overviewof quantum information theory as it relates to computation, discussingentanglement, quantum entropy, and information-theoretic limits like theHolevo bound. Finally, we review the leading physical implementations(quantum hardware models) of quantum computers—trapped ions, super-conducting circuits, photonic systems, solid-state spin qubits, and oth-ers—outlining the challenges and achievements of each approach.

Quantum systems are very delicate, both because they tend to be very small, and thus vulnerable to even tiny disturbances, and because any process that gains information about the state of a quantum system alters it. Nevertheless, quantum coherence can, in principle, be maintained indefinitely using error-correcting codes that shield quantum information from the ravages of a harsh environment. Protected quantum information in turn enables large-scale quantum technologies, such as quantum computers, which can take advantage of quantum interference to solve some computational problems much faster than any conceivable classical computer. One promising approach is to use topological error-correcting codes, which store quantum information safely by associating it with some topological property of the system, such as a path stretching all the way around it. Now, in a paper appearing in Physical Review X, Hector Bombin at the Perimeter Institute in Waterloo, Canada and an international group of scientists from Switzerland, Japan and the US have collaborated to study two families of topological codes and determine how much protection they provide against the most symmetric type of errors [1]. They do this by making a connection between the error-correcting codes and certain purely classical systems. The low-error regime, where the quantum code functions, corresponds to the low-temperature ordered phase in the classical system.

Surface code is an error-correcting method that can be applied to the implementation of a usable quantum computer. At present, a promising candidate for a usable quantum computer is based on superconductor-specifically transmon. Because errors in transmon-based quantum computers appear biasedly as Z type errors, tailored surface and XZZX codes have been developed to deal with the type errors. Even though these surface codes have been suggested for lattice structures, since transmons-based quantum computers, developed by IBM, have a heavy-hexagon structure, it is natural to ask how tailored surface code and XZZX code can be implemented on the heavy-hexagon structure. In this study, we provide a method for implementing tailored surface code and XZZX code on a heavy-hexagon structure. Even when there is no bias, we obtain $$ 0.231 \%$$ as the threshold of the tailored surface code, which is much better than $$ 0.21 \%$$ and $$ 0.209 \%$$ as the thresholds of the surface code and XZZX code, respectively. Furthermore, we can see that even though a decoder, which is not the best of the syndromes, is used, the thresholds of the tailored surface code and XZZX code increase as the bias of the Z error increases. Finally, we show that in the case of infinite bias, the threshold of the surface code is $$ 0.264\%$$ , but the thresholds of the tailored surface code and XZZX code are $$ 0.296 \% $$ and $$ 0.328 \%$$ respectively.

Circuit-based quantum computing (CBQC), also known as the gate model, is currently regarded as the leading paradigm for performing quantum computation and has garnered widespread interest due to its relatively better-established implementation and close intuitive resemblance to classical computation. However, an alternative model that solely utilizes single-qubit measurements on entangled states to perform computation, namely the measurement-based quantum computing (MBQC), has started to attract further research interest as it is deemed more naturally suited for certain quantum hardware architectures, though its actual implementation remains largely underexplored. Recently, there have been preliminary efforts to characterize the specific regimes in which MBQC may potentially be more advantageous compared to CBQC. Nevertheless, a more systematic assessment is necessary to assess the suitability of each model across a broader range of more practical cases. In this work, we present a detailed comparative analysis of CBQC and MBQC, beginning with reviewing the theoretical foundations of each model before highlighting its distinguishing features that set the two paradigms apart. Furthermore, building on previous work, we propose a generalized regime analysis that more accurately captures the conditions under which a given quantum algorithm is more favorably executed in MBQC or CBQC. Subsequently, we identify critical parameters that likely determine whether a quantum algorithm is better suited to CBQC or MBQC, categorizing detailed key factors and performance metrics into layers of quantum computation. Next, we propose a decision flow that, given specific hardware constraints and algorithm requirements, guides the selection to the most appropriate model, discussing suitability in two representative worked examples of quantum algorithm applications: Quantum Fourier Transform (QFT) and Variational Quantum Algorithm (VQE). Last, we discuss the potential to combine both models as a hybrid approach to leverage the best of both worlds and explore its benefits for practical quantum computing applications.

Measurement-Based Quantum Computing (MBQC), proposed in 2001 is a model of quantum computing that achieves quantum computation by performing a series of adaptive single-qubit measurements on an entangled cluster state. Our project is aimed at introducing MBQC to a wide audience ranging from high school students to quantum computing researchers through a Tangram puzzle with a modified set of rules played on an applet. The player is provided a quantum circuit which they have to map to MBQC using polyominos. Polyominos, the building blocks of our game, consist of square tiles joined edge-to-edge to form different shapes. Each tile represents a measurement basis, differentiated by its color. Polyominos rest on a square-grid playing board, which signifies a cluster state. We show that mapping a quantum circuit to MBQC is equivalent to arranging a set of polyominos—each corresponding to a gate in the circuit—on the playing board, subject to certain rules. We state the rules in simple terms with no reference to quantum computing. One such rule describes ways to deform a polyomino while it still correctly realizes a given quantum gate. The player has to place polyominos on the playing board conforming to the rules. Any correct solution creates a valid realization of the quantum circuit in MBQC. A higher-scoring correct solution fills up less space on the board, resulting in a lower-overhead embedding of the circuit in MBQC, a challenging research problem.

This paper presents a method for enumerating all encoding operators in the Clifford group for a given stabilizer. Furthermore, we classify encoding operators into the equivalence classes such that EDPs (Entanglement Distillation Protocol) constructed from encoding operators in the same equivalence class have the same performance. By this classification, for a given parameter, the number of candidates for good EDPs is significantly reduced. As a result, we find the best EDP among EDPs constructed from [[4,2]] stabilizer codes. This EDP has a better performance than previously known EDPs over wide range of fidelity.

Developing quantum systems which are robust against noise are of prime importance to the realisation of quantum technologies. Without fault tolerance, we will never be able to preserve delicate quantum states for macroscopic time scales, a necessary requirement in the construction of scalable quantum computer. Topologically ordered models offer very beautiful mechanisms for preserving quantum states. In such models, quantum information is encoded globally over a degenerate topological Hilbert space which offers a natural robustness against local environmental noise. Remarkably, topological order is present in certain realistic condensed-matter systems. This provides a platform for accessing topological order in a laboratory. Moreover, considering condensed-matter systems enables us to combine topological features with other physical effects to enhance their behaviour. In this Thesis we study the physics of topologically ordered lattice models with defects. We seek practical applications of lattice defects for the realisation of a fault-tolerant quantum computer. The first result we present shows that anyonic data of point-like lattice defects, called twists, can be found by measuring the entanglement of the ground state of the host system. The data we learn relates to the capacity of a twist defect to perform quantum computational tasks. The second result in this Thesis concerns the dynamics of the qudit toric code model with line-like defects coupled to a thermal bath. We show we can entropically inspire fragile glassy dynamics in the system. Such dynamics qualitatively improve the coherence times of quantum information encoded in the ground state of the lattice. A novel way of achieving fault-tolerant quantum computation is by producing and manipulating twist defects of topologically ordered systems. In many ways, this paradigm of topological quantum computation is analogous to quantum computation using anyons. The first area of study in this Thesis extends the analogy between anyons and twists. Specifically, we show that the anyonic data of twists in Kitaev’s toric code model can be extracted

Logic qubit plays an important role in quantum computation. Recent research showed that logic qubit entanglement can be used in long-distance quantum communication. In this paper, we describe an entanglement purification protocol (EPP) for one type of logic qubit entanglement, named concatenated Greenberger–Horne–Zeilinger(C-GHZ) state. Not only the logic bit-flip and phase-flip errors in the C-GHZ state but also the single physical qubit bit-flip and phase-flip errors in the logic qubit can be purified. This EPP does not require the sophisticated controlled-not gate, but only requires some feasible linear optical elements, which is feasible in the current experimental condition.

The field of quantum computation currently lacks a formal proof of experimental feasibility. Qubits are fragile and sophisticated quantum error correction is required to achieve reliable quantum computation. The surface code is a promising quantum error correction code, requiring only a physically reasonable 2D lattice of qubits with nearest neighbor interactions. However, existing proofs that reliable quantum computation is possible using this code assume the ability to measure four-body operators and, despite making this difficult to realize assumption, require that the error rate of these operator measurements is less than 10(-9), an unphysically low target. High error rates have been proved tolerable only when assuming tunable interactions of strength and error rate independent of distance, which is also unphysical. In this work, given a 2D lattice of qubits with only nearest neighbor two-qubit gates, and single-qubit measurement, initialization, and unitary gates, all of which have error rate p, we prove that arbitrarily reliable quantum computation is possible provided p < 7.4 × 10(-4), a target that many experiments have already achieved. This closes a long-standing open problem, formally proving the experimental feasibility of quantum computation under physically reasonable assumptions.

One of the most promising routes towards fault-tolerant quantum computation utilizes topological quantum error correcting codes, such as the $\mathbb{Z}_2$ surface code. Logical qubits can be encoded in a variety of ways in the surface code, based on either boundary defects, holes, or bulk twist defects. However proposed fault-tolerant implementations of the Clifford group in these schemes are limited and often require unnecessary overhead. For example, the Clifford phase gate in certain planar and hole encodings has been proposed to be implemented using costly state injection and distillation protocols. In this paper, we show that within any encoding scheme for the logical qubits, we can fault-tolerantly implement the full Clifford group by using joint measurements involving a single appropriately encoded logical ancilla. This allows us to provide new low overhead implementations of the full Clifford group in surface and color codes. It also provides the first proposed implementations of the full Clifford group in hyperbolic codes. We further use our methods to propose state-of-the art encoding schemes for small numbers of logical qubits; for example, for code distances $d = 3,5,7$, we propose a scheme using $60, 160, 308$ (respectively) physical data qubits, which allow for the full logical Clifford group to be implemented on two logical qubits. To our knowledge, this is the optimal proposal to date, and thus may be useful for demonstration of fault-tolerant logical gates in small near-term quantum computers.

<p indent="0mm">Recently, the achievement, entitled “lifetime enhancement for a logic qubit by bosonic encoding error correction”, has been selected as one of “The Ten Major Advances in Chinese Science 2023”. Here, we will introduce briefly the importance of this work. It is known that quantum states are fragile due to decoherence induced by environment. Error will happen on a qubit, resulting in continuously changes of the parameters for this qubit, such as amplitude and phase parameters. The aim of quantum error correction is to amend those errors by using various quantum error correction codes, in which a logic qubit can be represented by several physical qubits. It should be noted that the quantum error correction can be realized by correcting a discrete set of errors, such as bit-flip error, phase-flip error or both of them. On the other hand, types of errors occurred depend on specific platforms of quantum computation. For example, photon loss is more inclined to occur in photonic systems. In this case, the approach of bosonic encoding of quantum error correction is more efficient. The work “lifetime enhancement for a logic qubit by bosonic encoding error correction” is for such a scenario. The challenge of quantum error correction is that operation itself is not perfect which may incur new errors. So it is actually difficult to demonstrate that we can benefit from quantum error correction. The achievement of lifetime enhancement by bosonic encoding error correction succeeds in beating this break-even point. The experiments are performed by circuit quantum electrodynamics on a superconducting processor with a qubit and a coupled resonate cavity. The logic qubit is realized by the bosonic mode of the cavity, while the superconducting qubit plays the role of the ancillary qubit in error correction. The initial states are prepared by the superconducting qubit, and are transferred to the logic qubit in the cavity. Then the error correction can be performed based on detection of photon loss by the superconducting qubit. The results show that the lifetime of the logic qubit increases from <sc>694 μs</sc> to <sc>805 μs</sc> with 16% enhancement. Besides high quality devices, the success of this experiment depends on the feedback control to detect and correct the error of photon loss in a short time. The result of beating the break-even point of bosonic encoding error correction is for one logic qubit. In the near future, it is hopeful to implement logic gates, such as single-qubit rotation gate and two-qubit gate, on a couple of logic qubits by bosonic encoding. The importance of the results will depend on whether higher fidelity can be achieved compared with those of gates with physical qubits. In this way, the advantage of the logic qubits for quantum computation is presented. It will be significant that the approach of bosonic encoding with a superconducting processor is scalable to hundreds of logic qubits, which may take a few years to realize. Our aim is to realize a fault-tolerant universal quantum computer based on a large quantity of high-precision logic qubits. The work “lifetime enhancement for a logic qubit by bosonic encoding error correction” is an important step toward this aim.

The Google Quantum AI team's long-term goal is to realize a large-scale fault-tolerant quantum computer. Of the technologies available to implement the quantum processor at the core of such a system, solid-state superconducting circuits based on Josephson junctions (JJs) are among the strongest contenders. Recent demonstrations with this technology include computation beyond the capabilities of today's most powerful supercomputers [1] and execution of quantum error correction (QEC) codes that can detect and correct either bit or phase flip errors [2]. These exciting results were enabled by low-error-rate quantum processors with tens of qubits and the supporting infrastructure to run these devices. However, today's quantum computers still have orders of magnitude fewer qubits than is required for a useful fault-tolerant quantum computer; for this, an estimated million or more qubits will be needed and scaling-up today's technology to this extent is non-trivial. Here, we review the Google Quantum AI team's recent progress and future plans. Specifically, we describe our post-classical Sycamore quantum computer and discuss the Google Quantum AI roadmap to fault-tolerant quantum computing.

Although qubits are the leading candidate for the basic elements in a quantum computer, there are also a range of reasons to consider using higher dimensional qudits or quantum continuous variables (QCVs). In this paper we use a general `quantum variable' formalism to propose a method of quantum computation in which ancillas are used to mediate gates on a well-isolated `quantum memory' register and which may be applied to the setting of qubits, qudits (for $d>2$) or QCVs. More specifically, we present a model in which universal quantum computation may be implemented on a register using only: repeated applications of a single fixed two-body ancilla-register interaction gate, ancillas prepared in a single state, and local measurements of these ancillas. In order to maintain determinism in the computation, adaptive measurements via a classical-feedforward of measurement outcomes are used, with the method similar to that in measurement-based quantum computation (MBQC). We show that our model has the same hybrid quantum-classical processing advantages as MBQC, including the power to implement any Clifford circuit in essentially one layer of quantum computation. In some physical settings, high-quality measurements of the ancillas may be highly challenging or not possible, and hence we also present a globally unitary model which replaces the need for measurements of the ancillas with the requirement for ancillas to be prepared in states from a fixed orthonormal basis. Finally, we discuss settings in which these models may be of practical interest.

In measurement-based quantum computation (MBQC), a special highly-entangled state (called a resource state) allows for universal quantum computation driven by single-qubit measurements and post-measurement corrections. Physical realisations of this model have been achieved in various physical systems for low numbers of qubits. The large number of qubits necessary to construct the resource state constitutes one of the main down sides to MBQC. However, in some instances it is possible to extend the resource state on the fly, meaning that not every qubit must be realised in the devices simultaneously. We consider the question of the minimal number of physical qubits that must be present in a system to directly implement a given measurement pattern. For measurement patterns with n inputs, n outputs and m total qubits which have flow, we show that only min(n + 1, m) qubits are required, while the number of required qubits can be as high as m-2 for measurement patterns with only gflow. We discuss the implications of removing the Clifford part of a measurement pattern, using well-established transformation rules for Pauli measurements, for the presence of flow versus gflow, and hence the effect on the minimum number of physical qubits required to directly realise the measurement pattern.

Cluster states, a special type of highly entangled states, are a universal resource for measurement-based quantum computation. Here, we propose an efficient one-step generation scheme for cluster states in semiconductor quantum dot molecules, where qubits are encoded on singlet and triplet states of two coupled quantum dots. By applying a collective electrical field or simultaneously adjusting interdot bias voltages of all double-dot molecules, we get a switchable Ising-like interaction between any two adjacent quantum molecule qubits. The initialization, the single-qubit measurement, and the experimental parameters are discussed. It is shown that preparation of large cluster states and one-way quantum computation are implementable in semiconductor quantum dots with the present techniques.