Approximate programmable quantum processors

Estimation of the offset between two network clocks has received a lot of attention in the literature, with the motivating force being data networking applications that require synchronous communication protocols. Statistical modeling techniques have been used to develop improved estimation algorithms, with a recent development being the construction of a confidence interval based on a fixed sample size. Lacking in the fixed sample size confidence interval procedures is a useable relationship between sample size and the width of the resulting confidence interval. Were that available, an optimum sample size could be determined to achieve a specified level of precision in the estimator and thereby improve the efficiency of the estimation procedure by reducing unnecessary overhead in the network that is associated with collecting the data used by the estimation schemes. A fixed sample size confidence interval that has a prescribed width is not available for this problem. However, in this paper we develop and compare alternative sequential intervals with fixed width and demonstrate that an effective solution is available.

An algebraic method of strain analysis using elliptical markers

The paradigm of variational quantum classifiers (VQCs) encodes classical information as quantum states, followed by quantum processing and then measurements to generate classical predictions. VQCs are promising candidates for efficient utilizations of noisy intermediate scale quantum (NISQ) devices: classifiers involving M-dimensional datasets can be implemented with only lceil log _2 M rceil qubits by using an amplitude encoding. A general framework for designing and training VQCs, however, is lacking. An encouraging specific embodiment of VQCs, quantum circuit learning (QCL), utilizes an ansatz: a circuit with a predetermined circuit geometry and parametrized gates expressing a time-evolution unitary operator; training involves learning the gate parameters through a gradient-descent algorithm where the gradients themselves can be efficiently estimated by the quantum circuit. The representational power of QCL, however, depends strongly on the choice of the ansatz, as it limits the range of possible unitary operators that a VQC can search over. Equally importantly, the landscape of the optimization problem may have challenging properties such as barren plateaus and the associated gradient-descent algorithm may not find good local minima. Thus, it is critically important to estimate (i) the price of ansatz; that is, the gap between the performance of QCL and the performance of ansatz-independent VQCs, and (ii) the price of using quantum circuits as classical classifiers: that is, the performance gap between VQCs and equivalent classical classifiers. This paper develops a computational framework to address both these open problems. First, it shows that VQCs, including QCL, fit inside the well-known kernel method. Next it introduces a framework for efficiently designing ansatz-independent VQCs, which we call the unitary kernel method (UKM). The UKM framework enables one to estimate the first known computationally-determined bounds on both the price of ansatz and the price of any speedup advantages of VQCs: numerical results with datatsets of various dimensions, ranging from 4 to 256, show that the ansatz-induced gap can vary between 10 and 20%, while the VQC-induced gap (between VQC and kernel method) can vary between 10 and 16%. To further understand the role of ansatz in VQCs, we also propose a method of decomposing a given unitary operator into a quantum circuit, which we call the variational circuit realization (VCR): given any parameterized circuit block (as for example, used in QCL), it finds optimal parameters and the number of layers of the circuit block required to approximate any target unitary operator with a given precision.

The quantum switch is a quantum process that creates a coherent control between different unitary operations, which is often described as a quantum process which transforms a pair of unitary operations (U1,U2) into a controlled unitary operation that coherently applies them in different orders as |0⟩⟨0|⊗U1U2+|1⟩⟨1|⊗U2U1. This description, however, does not directly define its action on non-unitary operations. The action of the quantum switch on non-unitary operations is then chosen to be a ``natural'' extension of its action on unitary operations. In general, the action of a process on non-unitary operations is not uniquely determined by its action on unitary operations. It may be that there could be a set of inequivalent extensions of the quantum switch for non-unitary operations. We prove, however, that the natural extension is the only possibility for the quantum switch for the 2-slot case. In other words, contrary to the general case, the action of the quantum switch on non-unitary operations (as a linear and completely CP preserving supermap) is completely determined by its action on unitary operations. We also discuss the general problem of when the complete description of a quantum process is uniquely determined by its action on unitary operations and identify a set of single-slot processes which are completely defined by their action on unitary operations.

Quantum error correction is a set of methods to protect quantum information—that is, quantum states—from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Most work on quantum error correction has focused on systems of quantum bits, or qubits, which are two-level quantum systems. These can be physically realized by the states of a spin-1/2 particle, the polarization of a single photon, two distinguished levels of a trapped atom or ion, the current states of a microscopic superconducting loop, or many other physical systems. The most widely used codes are the stabilizer codes, which are closely related to classical linear codes. The code space is the joint +1 eigenspace of a set of commuting Pauli operators on n qubits, called stabilizer generators; the error syndrome is determined by measuring these operators, which allows errors to be diagnosed and corrected. A stabilizer code is characterized by three parameters [[n,k,d]], where n is the number of physical qubits, k is the number of encoded logical qubits, and d is the minimum distance of the code (the smallest number of simultaneous qubit errors that can transform one valid codeword into another). Every useful code has n>k; this physical redundancy is necessary to detect and correct errors without disturbing the logical state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design. Other approaches to error mitigation in quantum systems include decoherence-free subspaces, noiseless subsystems, and dynamical decoupling.

We present an operational reconstruction of the well-known two-to-one homomorphism between the groups SU(2) and SO(3), grounded in the physical description of quantum state preparation and evolution. Starting from the connection between vectors in three-dimensional physical space and quantum states of two-level systems, we investigate how reversible transformations–modeled as completely positive and trace-preserving maps–give rise to a correspondence between spatial rotations and unitary operations. Our approach reveals how this group-theoretic structure naturally emerges from physical constraints, particularly the preservation of purity and reversibility in quantum processes. Beyond its theoretical relevance, the construction offers a pedagogically accessible framework for introducing core ideas in quantum mechanics and symmetry groups, making the abstract correspondence between SU(2) and SO(3) tangible through experimentally meaningful procedures.

Quantum resource theory is perhaps the most revolutionary framework that quantum physics has ever experienced. It plays vigorous roles in unifying the quantification methods of a requisite quantum effect as wells as in identifying protocols that optimize its usefulness in a given application in areas ranging from quantum information to computation. Moreover, the resource theories have transmuted radical quantum phenomena like coherence, nonclassicality and entanglement from being just intriguing to being helpful in executing realistic thoughts. A general quantum resource theoretical framework relies on the method of categorization of all possible quantum states into two sets, namely, the free set and the resource set. Associated with the set of free states there is a number of free quantum operations emerging from the natural constraints attributed to the corresponding physical system. Then, the task of quantum resource theory is to discover possible aspects arising from the restricted set of operations as resources. Along with the rapid growth of various resource theories corresponding to standard harmonic oscillator quantum optical states, significant advancement has been expedited along the same direction for generalized quantum optical states. Generalized quantum optical framework strives to bring in several prosperous contemporary ideas including nonlinearity, -symmetric non-Hermitian theories, q-deformed bosonic systems, etc., to accomplish similar but elevated objectives of the standard quantum optics and information theories. In this article, we review the developments of nonclassical resource theories of different generalized quantum optical states and their usefulness in the context of quantum information theories.

Quantum teleportation is one of the most basic quantum protocols, which transfers an unknown quantum state from one location to another through local operation and classical communication by using shared quantum entanglement without physical transfer of the information carrier. And it has been widely used in various quantum information protocols such as entanglement swapping, quantum repeaters, quantum gate teleportation, quantum computation based on measurement, and quantum teleportation networks, which have important application value in quantum computation and quantum information. Quantum teleportation is a naturally bipartite process, in which an unknown quantum state can only be transmitted from one node to another. With the further development of quantum information research, it is necessary to transfer quantum states or quantum information among more and more nodes. Multipartite quantum protocols are expected to form fundamental components for larger-scale quantum communication and computation. A bipartite quantum teleportation should be extended to a multipartite protocol known as a quantum teleportation network. In this paper, a multifunctional quantum teleportation network is proposed theoretically. We first propose a special method of constructing four-partite quantum resources in continuous variables (CVs), and based on this, construct two different types of CV quantum teleportation networks. One type of network contains just one quantum teleportation process consisting of a sender, a receiver and two controllers. In this type of network, the unknown quantum state can be recovered at any other node according to the requirement after the measurement in the input node, which enriches the transfer direction and transfer mode of the unknown quantum state. And meanwhile, the two controllers can control the transfer of a quantum state from the sender to the receiver by restricting the sender and receiver’s access to their information, which makes the quantum teleportation network controllable. The other type of network has two quantum teleportation processes, each containing only a sender, a receiver and no controllers, which increases the number of quantum states that can be transmitted. Then we analyze the dependence of the fidelity of each quantum teleportation network on different physical parameters, and compare the characteristics, advantages and disadvantages among different types of quantum teleportation networks. The scheme for constructing a multifunctional quantum teleportation network in this paper shows some advantages, such as the greater number of quantum nodes, diversity of types, simple operation procedure. And all these advantages provide a broader application prospect for establishing larger and more complex quantum information networks in the future and quicken the pace of the application of quantum information.

This chapter presents state-of-the-art and future perspectives of quantum computing and communication. Timeline of relevant findings in quantum informatics, such as quantum algorithms, quantum cryptography protocols, and quantum computing models, is summarized. Mathematics of information representation with quantum states is presented. The quantum circuit and adiabatic models of quantum computation are outlined. The functionality, limitations, and security of the quantum key distribution (QKD) protocol is presented. Current implementations of quantum computers and principles of quantum programming are shortly described.

No-partial erasure of quantum information

Quantum information theory has provided the formal framework for describing how information is stored, transmitted, and transformed in physical quantum systems [Renes, 2022; Tomamichel, 2015; Wilde, 2013]. Its entropic formulations underpin our understanding of quantum computation, communication, and cryptography. Yet this theory traditionally treats all quantum operations as freely available, ignoring computational restrictions. In practice, however, any manipulation of quantum information must be performed by devices of bounded complexity and runtime. Capturing such realistic constraints requires extending quantum information theory to include computational efficiency as a fundamental component. This work takes a first step toward building a computational version of quantum information theory, one that treats efficiency as part of the theory itself. The goal is to understand how the behavior of quantum information changes when the parties involved can only perform computationally efficient operations. This approach bridges the abstract, ideal setting of quantum information theory with the practical limitations of real quantum devices, offering a means to study information processing under realistic resource constraints. At the center of this work are two new quantities: the quantum computational min-entropy and the quantum computational max-entropy. These entropies extend standard quantum entropies by explicitly limiting the computational power of the observer or adversary. The quantum computational min-entropy captures how unpredictable a quantum system A remains to an observer holding system B, when that observer is restricted to quantum circuits of bounded size. Formally, for a bipartite state ρ_{AB}, we define {H^c}^s_{min}(A|B)_{ρ} ≔ -log d_A max_{ℰ^s_{B→A'}} F((𝕀_A ⊗ ℰ^s)(ρ_{AB}),|Φ_{AA'}⟩⟨Φ_{AA'}|) , where the maximization is over quantum channels that can be implemented by circuits of size at most s, and F denotes fidelity with a maximally entangled state. In the classical setting, the min-entropy can be expressed through the maximal probability of correctly guessing a random variable given some side-information. In the fully quantum setting, this idea extends to uncertainty about quantum information [König et al., 2009], quantifying how well one system can be inferred from another using local quantum operations. Our definition generalizes this operational viewpoint by restricting the computational power of the observer to efficient quantum circuits. This definition extends the operational meaning of the information-theoretic quantum min-entropy [König et al., 2009] by incorporating computational constraints, and it provides the fully quantum counterpart of the classical unpredictability entropy [Hsiao et al., 2007]. We establish fundamental properties for the computational min-entropy, including monotonicity in the circuit size and smoothing parameters, efficient data-processing inequalities, and fully quantum leakage and purification chain rules, which were left as open questions in earlier definitions of quantum computational entropies [Yi-Hsiu Chen et al., 2017; Munson et al., 2025]. For classical–quantum states, it coincides with the previously defined quantum computational unpredictability entropy [Noam Avidan and Rotem Arnon, 2025], showing that the new definition correctly generalizes known results. We also introduce the quantum computational max-entropy through a duality relation [Tomamichel et al., 2010] with the min-entropy using a fixed purification. Finally, we prove unconditional separations between the computational and information-theoretic entropies, demonstrating that computational restrictions can fundamentally alter entropic behavior even for simple states. These results establish the fundamental mathematical framework for studying quantum information within realistic computational constraints. By integrating efficiency directly into entropic quantities, they open the door to a fully developed computational quantum information theory that parallels its information-theoretic counterpart. Such a framework provides the foundation for analyzing cryptographic security against computationally bounded quantum adversaries [Noam Avidan and Rotem Arnon, 2025] and the limits of efficient quantum state manipulation. More broadly, it suggests that many core notions in quantum information theory may have refined computational analogues yet to be explored.

In this paper we present a novel approach to emulating a universal quantum computer with a classical system, one that uses a signal of bounded duration and amplitude to represent an arbitrary quantum state. The signal may be of any modality (e.g., acoustic, electromagnetic, etc), but we focus our discussion here on electronic signals. Unitary gate operations are performed using analog electronic circuit devices, such as four-quadrant multipliers, operational amplifiers, and analog filters, although non-unitary operations may be performed as well. In this manner, the Hilbert space structure of the quantum state, as well as a universal set of gate operations, may be fully emulated classically. The required bandwidth scales exponentially with the number of qubits, however, thereby limiting the scalability of the approach, but the intrinsic parallelism, ease of construction, and classical robustness to decoherence may nevertheless lead to capabilities and efficiencies rivaling that of current high performance computers.

In order to investigate distributed quantum computation under restricted network resources, we introduce a quantum computation task over the butterfly network where both quantum and classical communications are limited. We consider deterministically performing a two-qubit global unitary operation on two unknown inputs given at different nodes, with outputs at two distinct nodes. By using a particular resource setting introduced by M. Hayashi [Phys. Rev. A \textbf{76}, 040301(R) (2007)], which is capable of performing a swap operation by adding two maximally entangled qubits (ebits) between the two input nodes, we show that unitary operations can be performed without adding any entanglement resource, if and only if the unitary operations are locally unitary equivalent to controlled unitary operations. Our protocol is optimal in the sense that the unitary operations cannot be implemented if we relax the specifications of any of the channels. We also construct protocols for performing controlled traceless unitary operations with a 1-ebit resource and for performing global Clifford operations with a 2-ebit resource.

Quantum error correction (QEC) is a way to protect quantum information against noise. It consists of encoding input information into entangled quantum states known as the code space. Furthermore, to classify if the encoded information is corrupted or intact, a step known as syndrome detection is performed. For stabilizer codes, this step consists of measuring a set of stabilizer operators. In this paper, inspired by the QEC approach, and specifically stabilizer codes, we propose a prototype quantum circuit model for classification of classical data. The core quantum circuit can be considered as a quantum perceptron where the classification is based on syndrome detection. In this proposal, a quantum perceptron is realized by one stabilizer as part of a stabilizer code, while a quantum neural network (QNN) layer is realized by a stabilizer code which consists of many stabilizers. The concatenation of stabilizer codes results in complex QNNs. The QNN is trained by performing measurements and optimization of a set of parametrized stabilizers. We demonstrate the concept numerically. In this paper we also consider the first challenge to most applications of quantum computers, including data classification, which is to load data into the memory of the quantum computer. This loading amounts to representing the data as a quantum state, i.e., quantum state preparation. An exact amplitude encoding algorithm requires a circuit of exponential depth. We introduce an alternative recursive algorithm which approximates amplitude encoding with only a polynomial number of elementary gates. We name it the recursive approximate-scheme algorithm.

Controlling the evolution of photonic quantum states is crucial for most quantum information processing and metrology tasks. Due to its importance, many mechanisms of quantum state evolution have been tested in detail and are well understood; however, the fundamental phase anomaly of evolving waves, called the Gouy phase, has had a limited number of studies in the context of elementary quantum states of light, especially in the case of photon number states. Here we outline a simple method for calculating the quantum state evolution upon propagation and demonstrate experimentally how this quantum Gouy phase affects two-photon quantum states. Our results show that the increased phase sensitivity of multi-photon states also extends to this fundamental phase anomaly and has to be taken into account to fully understand the state evolution. We further demonstrate how the Gouy phase can be used as a tool for manipulating quantum states of any bosonic system in future quantum technologies, outline a possible application in quantum-enhanced sensing, and dispel a common misconception attributing the increased phase sensitivity of multi-photon quantum states solely to an effective de Broglie wavelength.