Introduction to Quantum Computation and Information

A quantum computer is a powerful machine that provides a new approach to information processing based on quantum mechanics. However, basically, qubits are very vulnerable to errors, thus a quantum error correction code (QECC) is needed to build a reliable quantum computer. Moreover, simply using QECCs is not sufficient in order that a quantum computer can tolerate errors. In quantum computation, each operation must be performed on the encoded states. Even when the gates used to perform these operations are noisy, the quantum computation must be tolerable to the errors. This is called fault-tolerant quantum computation and it is an essential condition for implementing a quantum computer. However, considering the fault tolerance, the overhead of the quantum computer increases exponentially. In this paper, we introduce fault-tolerant quantum computation and schemes to reduce the its overhead.

This paper shows that quantum computation can be made fault-tolerant against errors and inaccuracies when $\eta$, the probability for an error in a qubit or a gate, is smaller than a constant threshold $\eta_c$. This result improves on Shor's result [Proceedings of the 37th Symposium on the Foundations of Computer Science, IEEE, Los Alamitos, CA, 1996, pp. 56–65], which shows how to perform fault-tolerant quantum computation when the error rate $\eta$ decays polylogarithmically with the size of the computation, an assumption which is physically unreasonable. The cost of making the quantum circuit fault-tolerant in our construction is polylogarithmic in time and space. Our result holds for a very general local noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space and in time. Fault-tolerant computation can be performed with any universal set of gates. The result also holds for quantum particles with $p>2$ states, namely, p-qudits, and is also generalized to one-dimensional quantum computers with only nearest-neighbor interactions. No measurements, or classical operations, are required during the quantum computation. We estimate the threshold of our construction to be $\eta_c\simeq 10^{-6}$, in the best case. By this we show that local noise is in principle not an obstacle for scalable quantum computation. The main ingredient of our proof is the computation on states encoded by a quantum error correcting code (QECC). To this end we introduce a special class of Calderbank–Shor–Steane (CSS) codes, called polynomial codes (the quantum analogue of Reed–Solomon codes). Their nice algebraic structure allows all of the encoded gates to be transversal. We also provide another version of the proof which uses more general CSS codes, but its encoded gates are slightly less elegant. To achieve fault tolerance, we encode the quantum circuit by another circuit by using one of these QECCs. This step is repeated polyloglog many times, each step slightly improving the effective error rate, to achieve the desired reliability. The resulting circuit exhibits a hierarchical structure, and for the analysis of its robustness we borrow terminology from Khalfin and Tsirelson [Found. Phys., 22 (1992), pp. 879–948] and Gács [Advances in Computing Research: A Research Annual: Randomness and Computation, JAI Press, Greenwich, CT, 1989]. The paper is to a large extent self-contained. In particular, we provide simpler proofs for many of the known results we use, such as the fact that it suffices to correct for bit-flips and phase-flips, the correctness of CSS codes, and the fact that two-qubit gates are universal, together with their extensions to higher-dimensional particles. We also provide full proofs of the universality of the sets of gates we use (the proof of universality was missing in Shor's paper). This paper thus provides a self-contained and complete proof of universal fault-tolerant quantum computation in the presence of local noise.

Continuous-variable measurement-based quantum computation, which requires deterministically generated large-scale cluster states, is a promising candidate for practical, scalable, universal, and fault-tolerant quantum computation. In this work, based on our compact and scalable scheme of generating a two-dimensional spatiotemporal cluster state, a complete architecture including cluster state preparation, gate implementations, and error correction, is proposed. First, a scheme for generating two-dimensional large-scale continuous-variable cluster state by multiplexing both the temporal and spatial domains is proposed. Then, the corresponding gate implementations by gate teleportation are discussed and the actual gate noise from the generated cluster state is considered. After that, the quantum error correction can be further achieved by utilizing the square-lattice Gottesman-Kitaev-Preskill (GKP) code. Finally, a fault-tolerant quantum computation can be realized by introducing bias into the square-lattice GKP code (to protect against phase-flip errors) and concatenating a repetition code (to handle the residual bit-flip errors), with a squeezing threshold of 12.3 dB. Our work provides a possible option for a complete fault-tolerant quantum computation architecture in the future.

One reason why building large-scale quantum computers will be particularly challenging is that quantum computers are far more susceptible to making errors than conventional digital computers. I will explain the principles of quantum error correction and fault-tolerant quantum computation, which can enable a properly designed quantum computer with imperfect components to achieve good reliability. Some recent developments in the theory of quantum fault tolerance will be reviewed, including quantum error-correcting codes for systems described by continuous quantum variables, robust storage of quantum information based on topological principles, and applications of quantum error-correcting codes to quantum cryptography.

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.

In a ground breaking discovery in 1994, Shor has shown that quantum computers, if built, can factor numbers efficiently. Since then quantum computing has become a burgeoning field of research, attracting theoreticians and experimentalists alike, and regrouping researchers from fields like computer science, physics, mathematics and engineering. Quantum information is very fragile and prone to decoherence. Yet by the middle of 1996 it has been shown that fault-tolerant quantum computation is possible. We give a simple description of the elements of quantum error-correction and quantum fault-tolerance. After characterizing quantum errors we present several error correction schemes and outline the elements of a full fledged fault-tolerant computation, which works error-free even though all of its components can be faulty. We also mention alternative approaches to error-correction, so called error-avoiding or decoherence-free schemes.

<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.

Compiling high-level quantum programs to machines that are size constrained (i.e. limited number of quantum bits) and time constrained (i.e. limited number of quantum operations) is challenging. In this paper, we present SQUARE (Strategic QUantum Ancilla REuse), a compilation infrastructure that tackles allocation and reclamation of scratch qubits (called ancilla) in modular quantum programs. At its core, SQUARE strategically performs uncomputation to create opportunities for qubit reuse. Current Noisy Intermediate-Scale Quantum (NISQ) computers and forward-looking Fault-Tolerant (FT) quantum computers have fundamentally different constraints such as data locality, instruction parallelism, and communication overhead. Our heuristic-based ancilla-reuse algorithm balances these considerations and fits computations into resource-constrained NISQ or FT quantum machines, throttling parallelism when necessary. To precisely capture the workload of a program, we propose an improved metric, the "active quantum volume," and use this metric to evaluate the effectiveness of our algorithm. Furthermore, our results show that SQUARE improves the average success rate of NISQ applications by 1.47X. Surprisingly, the additional gates for uncomputation create ancilla with better locality, and result in substantially fewer swap gates and less gate noise overall. SQUARE also achieves an average reduction of 1.5X (and up to 9.6X) in active quantum volume for FT machines.

During recent years, quantum computers have received increasing attention, primarily due to their ability to significantly increase computational performance for specific problems. Computational performance could be improved for mathematical optimization by quantum annealers. This special type of quantum computer can solve quadratic unconstrained binary optimization problems. However, multi-energy systems optimization commonly involves integer and continuous decision variables. Due to their mixed-integer problem structure, quantum annealers cannot be directly used for multi-energy system optimization.To solve multi-energy system optimization problems, we present a hybrid Benders decomposition approach combining optimization on quantum and classical computers. In our approach, the quantum computer solves the master problem, which involves only the integer variables from the original energy system optimization problem. The subproblem includes the continuous variables and is solved by a classical computer. For better performance, we apply improvement techniques to the Benders decomposition. We test the approach on a case study to design a cost-optimal multi-energy system. While we provide a proof of concept that our Benders decomposition approach is applicable for the design of multi-energy systems, the computational time is still higher than for approaches using classical computers only. We therefore estimate the potential improvement of our approach to be expected for larger and fault-tolerant quantum computers.

The seven-qubit quantum error-correcting code originally proposed by Steane is one of the best known quantum codes. The Steane code has a desirable property that most basic operations can be performed easily in a fault-tolerant manner. A major obstacle to fault-tolerant quantum computation with the Steane code is fault-tolerant preparation of encoded states, which requires large computational resources. Here we propose efficient state preparation methods for zero and magic states encoded with the Steane code, where the zero state is one of the computational basis states and the magic state allows us to achieve universality in fault-tolerant quantum computation. The methods minimize resource overheads for the fault-tolerant state preparation, and therefore reduce necessary resources for quantum computation with the Steane code. Thus, the present results will open a new possibility for efficient fault-tolerant quantum computation.

To implement fault-tolerant quantum computation with continuous variables, the Gottesman--Kitaev--Preskill (GKP) qubit has been recognized as an important technological element. We have proposed a method to reduce the required squeezing level to realize large scale quantum computation with the GKP qubit [Phys. Rev. X. {\bf 8}, 021054 (2018)], harnessing the virtue of analog information in the GKP qubits. In the present work, to reduce the number of qubits required for large scale quantum computation, we propose the tracking quantum error correction, where the logical-qubit level quantum error correction is partially substituted by the single-qubit level quantum error correction. In the proposed method, the analog quantum error correction is utilized to make the performances of the single-qubit level quantum error correction almost identical to those of the logical-qubit level quantum error correction in a practical noise level. The numerical results show that the proposed tracking quantum error correction reduces the number of qubits during a quantum error correction process by the reduction rate $\left\{{2(n-1)\times4^{l-1}-n+1}\right\}/({2n \times 4^{l-1}})$ for $n$-cycles of the quantum error correction process using the Knill's $C_{4}/C_{6}$ code with the concatenation level $l$. Hence, the proposed tracking quantum error correction has great advantage in reducing the required number of physical qubits, and will open a new way to bring up advantage of the GKP qubits in practical quantum computation.

Fault-tolerant quantum computers that can solve hard problems rely on quantum error correction1. One of the most promising error correction codes is the surface code2, which requires universal gate fidelities exceeding an error correction threshold of 99 per cent3. Among the many qubit platforms, only superconducting circuits4, trapped ions5 and nitrogen-vacancy centres in diamond6 have delivered this requirement. Electron spin qubits in silicon7-15 are particularly promising for a large-scale quantum computer owing to their nanofabrication capability, but the two-qubit gate fidelity has been limited to 98 per cent owing to the slow operation16. Here we demonstrate a two-qubit gate fidelity of 99.5 per cent, along with single-qubit gate fidelities of 99.8 per cent, in silicon spin qubits by fast electrical control using a micromagnet-induced gradient field and a tunable two-qubit coupling. We identify the qubit rotation speed and coupling strength where we robustly achieve high-fidelity gates. We realize Deutsch-Jozsa and Grover search algorithms with high success rates using our universal gate set. Our results demonstrate universal gate fidelity beyond the fault-tolerance threshold and may enable scalable silicon quantum computers.

This thesis presents the development of resource-efficient photon-distillation schemes designed to produce highly indistinguishable photons, a key requirement for large-scale photonic quantum computing. Although photons do not naturally interact, their indistinguishability enables effective interactions through quantum interference, allowing entanglement generation using only linear optical elements and measurements. In practice, however, photons are never perfectly identical. Small differences introduce decoherence, limiting the scalability and reliability of photonic quantum technologies. Addressing indistinguishability errors is therefore essential for fault-tolerant quantum computing.<br/><br/>The thesis first reviews the importance of photon indistinguishability and the need to mitigate related errors. It then examines experimental methods for characterizing imperfections and introduces a general model of imperfect photon states. This model captures indistinguishability errors alongside additional experimental noise from photon loss and multiphoton events. A key insight is that loss and multiphoton noise together can act as an effective source of indistinguishability error. To quantify this effect, the work defines an effective indistinguishability metric, enabling practical evaluation of photon-distillation schemes under realistic conditions.<br/><br/>The core theoretical contribution is the development of photon-distillation protocols based on multiphoton Fourier interference. Unlike earlier approaches, these schemes can asymptotically suppress indistinguishability errors to arbitrary levels while requiring only linear scaling in photon resources relative to the desired error reduction. The analysis further suggests that photon distillation, when combined with quantum error correction, can significantly reduce the overall resource overhead for fault-tolerant photonic quantum computing.<br/><br/>Experimentally, a three-photon Fourier-based distillation protocol was demonstrated on a high-fidelity photonic processor using a resonantly driven quantum-dot single-photon source. After subtracting excess noise contributions, a 2.2-fold reduction in indistinguishability error was achieved; when excess noise was retained, a 1.2-fold net reduction was observed. These results confirm a clear below-threshold improvement in photon quality. Additional analysis suggests that optimized distillation schemes could reduce the photon cost of a logical qubit by up to a factor of four. The thesis concludes by addressing insertion loss and outlining design strategies and future directions for integrating photon-distillation components into scalable quantum architectures.

Demonstrating small error rates by integrating quantum error correction (QEC) into an architecture of quantum computing is the next milestone towards scalable fault-tolerant quantum computing (FTQC). Encoding logical qubits with superconducting qubits and surface codes is considered a promising candidate for FTQC architectures. In this paper, we propose an FTQC architecture, which we call Q3DE, that enhances the tolerance to multi-bit burst errors (MBBEs) by cosmic rays with moderate changes and overhead. There are three core components in Q3DE: in-situ anomaly DEtection, dynamic code DEformation, and optimized error DEcoding. In this architecture, MBBEs are detected only from syndrome values for error correction. The effect of MBBEs is immediately mitigated by dynamically increasing the encoding level of logical qubits and re-estimating probable recovery operation with the rollback of the decoding process. We investigate the performance and overhead of the Q3DE architecture with quantum-error simulators and demonstrate that Q3DE effectively reduces the period of MBBEs by 1000 times and halves the size of their region. Therefore, Q3DE significantly relaxes the requirement of qubit density and qubit chip size to realize FTQC. Our scheme is versatile for mitigating MBBEs, i.e., temporal variations of error properties, on a wide range of physical devices and FTQC architectures since it relies only on the standard features of topological stabilizer codes.

In Chapter 1 we saw that open quantum systems could interact with an environment and that this coupling could turn pure states into mixed states. This process will detrimentally impact any quantum computation, because it can lessen or destroy the interference effects that are vital to distinguishing a quantum computer from a classical computer. The problem of overcoming this effect is called the decoherence problem . Historically, the problem of overcoming decoherence was thought to be a major obstacle towards building a quantum computer. However, it was discovered that, under suitable conditions, the decoherence problem could be overcome. The main idea behind how this can be achieved is through the theory of quantum error correction (QEC). In this chapter we give an introduction into the way in which the decoherence problem can be overcome via the method of QEC. It is important to note that the scope of the introduction is not comprehensive, and focuses only on the basics of QEC without reference to the notion of fault-tolerant quantum computation, which is covered in Chapter 5. Quantum error correction should be thought of as a (major) tool in this larger theory of fault-tolerant quantum computing.