Noise management to achieve superiority in quantum information systems

Abstract

Quantum information systems are expected to exhibit superiority compared with their classical counterparts. This superiority arises from the quantum coherences present in these quantum systems, which are obviously absent in classical ones. To exploit such quantum coherences, it is essential to control the phase information in the quantum state. The phase is analogue in nature, rather than binary. This makes quantum information technology fundamentally different from our classical digital information technology. In this paper, we analyse error sources and illustrate how these errors must be managed for the system to achieve the required fidelity and a quantum superiority.

This article is part of the themed issue ‘Quantum technology for the 21st century’.

1. Introduction

There have been many applications proposed to exploit ‘quantum coherence’ in quantum systems for technological advantages. These applications are spread over the entire information and computation technology (ICT) spectrum but can be broken down into three basic elements: information generation (data creation) [1,2], information transfer (data communication) [3–5] and information processing (data manipulation) [6]. In this regard, quantum metrology, sensing, imaging and high-precision measurements are ways to generate data [1,2], while quantum communication carries that data between locations (whether this be on the microscopic, macroscopic or global scale) [7–11]. Quantum computation is one way to manipulate data for quantum information processing [12]. In all of these cases, it is possible to realize technologies that classical ICT cannot provide.

Currently there is a large worldwide effort to design, demonstrate and realize quantum information technologies. On the basis of results from theory and experiment over the last two decades, we are now at the stage where we are seeing the emergence of small-scale quantum information technologies [13–15]. The advances in these new developments also apply and stimulate the conventional ICT industries, which certainly feeds back to the further developments of quantum information technology. This positive cycle of development is essential for the future ICT. We, however, need to stress that quantum technologies based on quantum coherence are not simple extensions of our current classical technologies. The main difference is that the advantages of quantum information technologies come from quantum coherence, a resource very different from what is used in classical technology. The manipulation and exploitation of quantum coherence for technologies have brought new concepts unique to quantum information technologies. For instance, a single electron transistor manipulates a single electron. In the conventional classical information processing we know the single spin carries only two values, ‘up’ and ‘down’; however in the quantum information world, the spin state can be represented as α|0〉+β|1〉, where α and β are complex numbers satisfying |α|²+|β|²=1 and include phase information [12]. In a sense, the quantum state carries analogue information (rather than discrete binary information) which needs to be maintained and manipulated. Owing to this nature, quantum information technology cannot simply be built on our existing technology and a paradigm change is needed [16,17].

Any technology that relies on quantum coherence (including phase information) requires a mechanism to overcome noise that will affect it. Although, as mentioned above, quantum information has a continuous nature due to how such information is represented on quantum states, we know that we can apply error correction codes and achieve a fault-tolerant implementation in theory, and the concept of error correction has been experimentally tested with various error correction codes. Quantum error correction codes digitize errors to perform error detection and correction [18]. Through the error correction procedure, the quantum information is kept as a continuous value, though errors are discretized into bit and/or phase errors [19,20]. This concept of quantum error correction brings features different from ones known in classical error correction, even though the construction of quantum error correction codes sometimes, though not always, can be considered to be a quantum extension of well-known classical error correction codes. Physical qubits are directly bound by our ability to manipulate the real material systems which construct them, and of course this restriction will be inherited by the respective logical qubits; however, the logical qubits are further bounded by the error correction protocols. Hence, how information is processed in the logical level may be governed by slightly different rules. A single qubit operation on a piece of encoded information will enlighten this difference. We are generally capable of performing arbitrary rotations on a single qubit (an extremely useful operation at the physical level) [12,21,22], but in the logical level, such arbitrary physical gate operations are not allowed and so not used as they do not maintain the code space [20,23]. In general, only several fixed angle single qubit operations that maintain the code space can be used. Thus, the inherent and useful capabilities present in the physical level could be completely irrelevant to quantum information processing in the logical level. In this paper, we focus on this fundamental difference between quantum and classical ICT and identify the elements required for us to scale quantum information technology up in a robust and efficient manner.

The continuous nature of information associated with quantum coherence in conjunction with quantum measurement makes the construction of quantum information technologies fundamentally different from the classical world, even at the theoretical level. While imperfections in a physical system and finite system control are common issues present in any technology, the effects and behaviour of errors in quantum information systems are rather unique. We now partially know how these errors occurring in our quantum information system affect the total system through recent research into quantum computing and communication architectures [6]. This has helped enlighten such unique features in quantum information systems. In this paper, we take various concepts that have been introduced in quantum architecture research and formulate these into a platform to manage errors. This is essential for quantum information technologies to achieve the promised advantages. It is interesting to note that, as nature does not usually reveal its quantum mechanical behaviour, the controllability of quantum information is not only the key for implementing quantum information technologies, but it also provides a guide to understand the complexity of quantum dynamics.

There are two main approaches to build a large general-purpose quantum information system (one independent of a particular application). First, we could design the total system as a single device manipulating a large number of physical parameters. Such an approach is usually taken for a small-scale system and/or devices, but it is not ideal at large scale simply because, as the system gets larger, its complexity gets too high to theoretically track or experimentally test. Even a system of tens of qubits can be too complex to be fully analysed, and so the majority of quantum information systems might soon be categorized as a large-scale system. For truly large systems, it should be better to take a divide-and-conquer approach where one breaks the larger system into a number of smaller pieces that can be connected together. We know from the recent quantum computing architecture work that we can assemble a system from building blocks which are small simple quantum devices with fixed functions [24–27]. The complexity and diversity of the systems will emerge from how we put them together rather than redesigning the building blocks for each application. Regardless, with both approaches, the errors at the physical level affect the fidelity of the information at the application level. However, with the latter approach, the error profile for each building block is independent and does not change as the system scales up. Hence, this model should more readily adapt to a layered error management [28] which we discuss in this paper.

2. Decoherence in quantum information systems

There have been quite a number of error correction, suppression and avoidance techniques proposed [29–34]. Some of them, such as spin echo [35–37] and pulse optimization [38–40], are well-established techniques in quantum physics, and some are more theoretical and difficult to implement, yet essential to realize a large-scale quantum information system. To build a quantum information system, we need to integrate several techniques to achieve the fidelity of quantum information for the application of our choice [24,28].

Now to avoid potential confusion, we will refer to the total system at hand as the quantum information system or simply the ‘system’ when we are interested in the application level (or layer). Hence, quantum information in the system carries information for the application and so any loss of information in this layer ends up as information loss in our results. When our focus is on a physical system, the state of the physical system suffers from errors, which accumulate in time, and these errors result in the decoherence of the physical system. However, decoherence of the individual physical system does not directly cause loss of information at the application level. Loss and errors can be handled in many different ways. Off-line resources are a typical example in our physical system (our physical system refers to the physical elements or hardware components in the quantum information system). Off-line qubits do not carry any application-level information. To illustrate the difference between the quantum information system and physical system, a quantum computer is the quantum information system while each qubit is the physical system.

In any quantum information system, such as quantum computers and quantum networks, for the system to be scalable, quantum error correction is absolutely necessary in some form. Consider that our system yields a certain error in a unit of time t. The total errors in the system during the entire run time will then accumulate. As there are always imperfections in any physical system, even when we suppress them to the theoretical limit, without a technique to correct errors on the application level, we expect the information on the application level will degrade over time. Unless there are absolutely no errors in the physical system, or the task at hand is completed before the total error reaches the critical point, the system always suffers from errors and requires error protection at that level to function as a system. It is easiest to take a top-down approach and begin with the application layer, but first let us define more explicitly what we mean by the application and physical layers. We refer to the error control on the system/application level as the application layer and the error control in the physical system as the physical layer (see figure 1). As we mentioned in the Introduction, the nature of manipulation in these two layers is fundamentally different, and so it is natural to divide into these two layers. The physical layer consists of several sublayers from the physical foundation of the devices. We first focus on the application and discuss the physical layer in the next subsection.

Figure 1.

Error and loss management in a large-scale quantum device. The device is conceptionally divided into a physical layer and an application layer. The physical layer is associated with the physical system and the tools necessary to manipulate the qubits present in it. The application layer runs ‘programs’ for the large-scale quantum device.

(a). The application layer

It is easiest to take a top-down approach and begin with the application layer. At the application level, we can consider two types of applications: applications with and without error suppression mechanism. For instance, adiabatic quantum computation is designed to perform computation by changing the Hamiltonian in time, and the slower the Hamiltonian changes, the less the error occurs in the system, that is, the more likely the computation yields the correct answer. This property could suggest that such implementation of quantum computation can be more stable to perturbation of the Hamiltonian in a comparison to the gate-model quantum computation where the gate errors will directly reflect to errors in the system. However, there exists so far no such error suppression mechanism that covers all the imperfection in the system. Once an error occurs at the application level, it stays in the system and accumulates in time. Hence, although error suppression can be done, it is always necessary to have an additional layer to control errors in the system. This usually can be dealt with by error correction codes.

A natural question to ask is when is it acceptable to omit using error correction codes at the application level? If we aim to implement a fault-tolerant system, no matter how small the system is, we do not have a choice: we always need to apply error correction codes in the fault-tolerant manner. However, there may be existing applications which do not need to be implemented fault-tolerantly. When the run time is fixed, we may not need to implement the system fault-tolerantly. However, a fault-tolerant implementation makes sure errors do not cascade via gates through the entire system. This in turn means the system purity can be high; however the total number of gates we can apply may be very limited. Quantum communications with a certain maximum distance could be one of these applications which could run without fault tolerance. The communication distance is usually limited based on the geometrical constraints on the Earth, and hence the running time can be fixed. The number of gate operations is determined based on the communication distance, hence an implementation of the system to guarantee the necessary fidelity for the distance is enough. Quantum computation and simulation, however, are not that straightforward to assess their viability in the non-fault-tolerant regime. When the problem at hand becomes harder to solve, the computational run time, i.e. the depth of the computation, gets longer. In the first approximation, a harder and larger problem requires a larger number of gates to complete the task. To achieve the fixed target fidelity, only a limited fixed number of gates can be applied. Now, we can also apply error correction codes in a non-fault-tolerant manner. This could extend the size of the computation we can run within the fixed system fidelity; however without the fault-tolerant implementation, errors could cascade through the system, generating more errors. It would be too optimistic to assume the substantial region where quantum computation and simulation could operate with high fidelity at the logical level without fault-tolerant implementation. The viable regime in a non-fault-tolerant implementation should be assessed with the details of the implementation of both the system and the application.

(b). The physical layer

Now we turn our attention to the physical layer and its sublayers as we illustrate in figure 1. To discuss this physical layer, it is easier to start it from the material side lowest sublayer and move towards the application layer. This is a bottom-up approach and so now let us go through each sublayer.

Environment design layer—The environment design is to show the physical structure of the fundamental building block. As this is the physical set-up of the building block, we cannot change it over time. Given this layer is to support the operations on it, its design is very much dependent on the physical layers above it. For instance, the physical design of the building blocks to accommodate control signals needed to realize building block functions. We need to choose the material, the energy levels, the environment conditions and so on to achieve high-fidelity device operation. When we control the environment, such as magnetic/electronic field strengths, strain in the material, etc., the physical set-up must adapt to this change to make the environment control more effective and stable.

Quantum manipulation layer—This layer controls all the building blocks. We need to perform operations such as initialization, gate operations and measurements within it. To do this, we need to manipulate the state of the physical systems, and it is necessary to design the manipulation to achieve the best possible fidelities.

Quantum environment control layer—This layer is here to actively control the environment of the fundamental building blocks. We control the environment both directly and indirectly. Spin echo is a typical example of indirect control as it corrects errors arising from classical incoherence (incoherence among the classical parameters). Though each spin rotates quantum coherently, the loss of the classical parameter, i.e. the rotation rate, causes an apparent de-phasing. One important and useful error control in this layer is the correction of errors due to classical incoherence.

Quantum protocol layer—This layer sits on top of the physical sublayers and can facilitate the use of off-line resources, which can be used to deal with higher error rates than the traditional error correction codes. This layer can deal with both such excess error rates and residual errors from the layers below.

Each layer has different error targets and hence can be designed separately. There is also a direction for which sublayer is more effective for errors in the physical system; however, there is still room to accommodate these layers into the design of the building blocks. In particular, the optimization of the error management cannot be efficiently done at each sublayer. Such an approach will create local optima, which will not work efficiently for the total system.

3. The quantum module and its architecture

It is useful to illustrate how these error controls can be applied to fault-tolerant quantum computation by examining an NV-based module approach as the fundamental building block (the module is illustrated in figure 2). The computational system can be assembled from these modules using quantum channels to connect them. Other elements required are single photon sources and detectors. Following the architecture shown in [24], we can now analyse how errors can be confined in a way that allows one to maintain the fidelity in the system level. Starting at the application level, we need to assume that all gates are implemented in a fault-tolerant manner for large-scale quantum computation. To do this, we define a logical qubit in a three-dimensional cluster state using the structures defined by three-dimensional topological quantum computation [41,42]. Fault-tolerant quantum error correction corrects errors at the application level and sets the target gate error rate based on the error threshold the architecture of the three-dimensional topological quantum computation gives [43] and the performance of classical error decoders used [44–46].

Figure 2.

(a) Illustration of the configuration of a module composed of a single negatively charged NV (nitrogen vacancy) centre embedded in an optical cavity. The operation of the module is based on dipole induced transparency. The cavity is tuned to reflect light when the NV electron spin state is in |0〉 and transmit it when it is in the |+1〉 state. Thus, the path the light travels is dependent on the state of the electron spin. The module also contains a nuclear spin (nitrogen-15) which serves as a long coherence time memory. The nuclear spin couples to the electron spin via a hyperfine coupling. (b) We show an entanglement distribution scheme based on this module concept using photon interference at a beamsplitter. The electron spin in each module is initialized as an equal superposition state of |0〉 and |+1〉. When the single photon is detected at the dark port, the gate has succeeded. Otherwise it will be repeated until success.

At the physical level, we need to provide the three-dimensional cluster state with a certain fidelity to achieve the target gate rate. The target error rate needs to be higher than the error rate after applying four layers of error management. We can start from any layer in the physical layer; we usually tend to start with the design of the physical system, i.e. the environment design layer, as we usually think based on a certain physical system at hand. However, in this paper, we start from the top layer, i.e. the quantum protocol layer; this approach will allow us to see the construction from the viewpoint of error management:

• — The quantum protocol layer is to absorb excess errors in the physical system. In our model of quantum computation, we have an optical channel to distribute entanglement between modules. Optical components today are not reliable at the single photon level, with both loss and inefficiency degrading the gate fidelity. Although there have been many improvements in photon detectors, on-demand single photon sources are still quite challenging, and further it is unlikely we will have zero (or extremely low) loss fibres/waveguides in the near future. A photon-mediated gate hence always suffers from imperfections far above the error threshold. Hence, it is impossible to deal with them only with the bottom three layers. In our case, we can perform the photon–electron gate off-line and use a post-selection approach. As the success of the gate is heralded, we can select the gate successful events and discard others without affecting the quantum computational data. Another example of error reduction using the quantum protocol is purification. When a gate is noisy, we can introduce post-selection in such a way as to gain the required gate fidelity by sacrificing the success probability. In this model of quantum computation, this function of purification can be used to accommodate imbalances from different cavities in the system, as they will not all be identical. Even if the technology allows us to realize each module to work with very low errors, it is impossible for each module to be exactly the same in reality. Such imbalances among cavities could degrade the gate fidelity; however, it is possible to design the scheme to trade off the success probability of the gate against the gate fidelity. The main target to control in this layer is the large noise, which is far above the threshold error rate, the excess noise in the physical layer, which could not be fully controlled in the bottom layers, and the additional noise emerging from the system, not individual modules. We should note that lowering the success probability requires a longer coherence time on qubits, and the other three layers need to deal with the tighter conditions for qubit coherence time. This illustrates how error management on each layer can be inter-related in the design of the building module and its operations.

• — The second top layer is to actively control the environment to suppress errors. Spin echo and dynamical decoupling can be thought of as typical examples. It changes the effective environment of the physical qubits in such a way that they show less decoherence. This is an additional layer and, if the environment is good enough for the physical qubits throughout operations, we do not need to apply this layer.

• — The next layer, the quantum manipulation layer, is associated with the active manipulation of the module. For instance, the electron bit-flip can be implemented by a microwave pulse, and the pulse should be optimized. This also applies to the connection between the modules, i.e. single photons need to be shaped so that the module can be operative with high fidelity. Hence, such optimization applies to all quantum and classical signals and parameters for active control of the physical system.

• — The environment design layer is about all the building blocks and how to minimize the imperfection of the system and its effects on the states. Hence, this layer is most discussed in realization of quantum information devices. Figure 2 shows the design of the device and how it operates. This layer is to realize the physical system to support all the quantum information processes in the system. The main task here is to minimize the errors that originate in this layer. If we can suppress enough in this layer and the quantum control layer, we might omit the quantum environment control layer in theory; however, it is usually difficult to do so, and we can see the reasons through this example.

The electron spin in the module is to interact with photons coming to the module (see figure 2) and hence it functions as an interface, whereas the nuclear spin is designed to carry the quantum computational data. The interface qubits and the quantum computational data qubits are assigned to physically different qubits, because any optical connection cannot meet the fidelity fault-tolerant error threshold without post-selection. First, we focus on the electron spin and identify the main error sources. De-phasing could be caused by the environment of the NV (nitrogen vacancy) centre. Hence, it is important to manufacture the NV centre to have long de-phasing time; for instance the magnetic and electric fields around the NV centre need to be stable. Depopulation of the states could also happen. As the energy difference is used to control the state of the electron, this type of error could be caused involving other energy states of the NV electron spin. As we see in figure 3, there can be unwanted transitions. In the future, we might be able to create and facilitate a material not to have metastable states; however it is too optimistic to think that we can control the branching ratio arbitrarily precisely by material and cavity design. Also, defects have unique characters; in the case of NV centres, each NV centre is slightly different, and the quantum properties in the ground manifold are different from the properties in the excited states. One might need to consider phonon-associated properties. Also, a population leakage needs to be considered, as well as the de-ionization of the NV centre. Although the rate for de-ionization is very small, errors which cannot be treated by the error correction code with the standard measurement schemes should not be neglected.

Figure 3.

Energy level diagram of the NV centre involving a ground and excited state manifold. The negatively charged NV centre exhibits good quantum properties, but it is not perfect in terms of the module design. The ground state manifold contains three energy levels from which two are chosen to define the qubit. The extra level (state) can cause leakage errors. Furthermore, the excited state manifold is formed from six energy levels (of which we only want to select two). The remaining four levels can also cause leakage or dephasing errors. Finally, the presence of metastable energy levels can also be an error source. The slow decay via metastable states results in a dead time in the module and a bit-flip error and a leakage error can happen in the decay process. The branching ratio can be controlled; however, it is hard to make existing channels to be effectively zero transition.

In our model of quantum computation, the nuclear spin is where the data are stored and processed. We can design the Hamiltonian in a way to help error correction become more efficient. The magnetic field is applied to control the coupling between the electron spin and the nuclear spin. This coupling consists of two parts: an energy exchange term and a ZZ conditional phase shift term. The energy exchange coupling is harder to correct and directly causes decoherence. On the other hand, the ZZ conditional phase shift coupling may lead to an unwanted phase shift, but techniques like spin echo can be used to correct this. The ZZ coupling can be more tolerant to certain errors. In this model, the electron spin will decohere when photon loss occurs but this will not affect the nuclear spin due to the ZZ coupling. Optical cavities are also an important element. To meet the target error rate, it is necessary to optimize the cavity design to achieve both fidelity and feasibility. Putting this all together we get an error and management layer diagram as depicted in figure 4.

Figure 4.

Error and loss management in an NV centre module-based large-scale quantum device.

4. Discussion and outlook

When we consider information processing in larger-scale quantum devices, it is useful to divide that process into two fundamental layers: a physical device layer responsible for control and manipulation of the ‘qubits’, and an application level layer to perform the computation/data manipulation (this is illustrated in figure 4 for an NV modular device). The physical layer can further be divided into a number of sublayers, each responsible for a different part of the computational system. Nonetheless, these sublayers are not completely independent of each other due to the fact that we cannot control errors in the computational system with only one sublayer. This complicates the system’s error management; however, it also gives us the opportunity to manage errors even when the physical system has weaknesses. Sublayers can work together to compensate weakness in the physical system (as illustrated in our previous example). The interaction between sublayers also complicates our attempt to optimize the overall system. Optimization of one sublayer may be detrimental to another sublayer. This fact was illustrated in our example when for instance a one-way cavity design was considered. The optimization needs to be done on the system, which clearly indicates the importance of the architecture of the system.

In this paper, we focused our attention on quantum information systems where we can realize large pure quantum states (large enough to run quantum information tasks). This is a necessary approach for these applications which require high-fidelity quantum coherence. However, there might be some class of quantum emulation which will not require such high quantum coherence. For instance, many quantum phenomena can be observed in a rather noisy environment. To understand these quantum phenomena, simulation of the Hamiltonian of the system might not be realistic. As real physical experiments involve the environment, the total model of the experiments would be even harder to analysis on a quantum computer. Quantum emulation in a noisy environment may be effective to extract important information in a very short period of time in comparison to the simulation time on both quantum and classical computers; however, it is not correct to suggest that such an approach can be a route to realize a quantum information system.

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Authors' contributions

K.N. conceived of the initial idea, and all authors contributed to the final design and the analysis. All authors contributed to the writing of the manuscript and gave final approval for publication.

Competing interests

The authors declare that there are no competing interests.

Funding

K.N. acknowledges support from the Japanese MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas Science of hybrid quantum systems (grant no. 15H05870).