Quantum Polynomial Research Articles

Error mitigation of BQP computations using measurement-based verification

We present a modular error mitigation protocol for running bounded-error quantum polynomial time (BQP) computations on a quantum computer with time-dependent noise. Utilizing existing tools from quantum verification and measurement-based quantum computation, our protocol interleaves standard computation rounds alongside test rounds for noise sampling and inherits an exponential bound (in the number of circuit runs) on the probability that a returned classical output is correct. We introduce a postselection technique called to address time-dependent noise and reduce overhead. The result is an error mitigation protocol which requires minimal noise assumptions, making it straightforwardly implementable on existing, noisy intermediate-scale quantum devices. We perform a demonstration of the protocol using classical noisy simulation, presenting a universal measurement pattern which directly maps to (and can be tiled on) the heavy-hex layout of current IBM hardware. Published by the American Physical Society 2025

Rewindable Quantum Computation and Its Equivalence to Cloning and Adaptive Postselection

We define rewinding operators that invert quantum measurements. Then, we define complexity classes RwBQP, CBQP, and AdPostBQP as sets of decision problems solvable by polynomial-size quantum circuits with a polynomial number of rewinding operators, cloning operators, and adaptive postselections, respectively. Our main result is that BPPPP⊆RwBQP=CBQP=AdPostBQP⊆PSPACE. As a byproduct of this result, we show that any problem in PostBQP can be solved with only postselections of events that occur with probabilities polynomially close to one. Under the strongly believed assumption that BQP⊉SZK, or the shortest independent vectors problem cannot be efficiently solved with quantum computers, we also show that a single rewinding operator is sufficient to achieve tasks that are intractable for quantum computation. Finally, we show that rewindable Clifford circuits remain classically simulatable, but rewindable instantaneous quantum polynomial time circuits can solve any problem in PP.

Classically simulating intermediate-scale instantaneous quantum polynomial circuits through a random graph approach

Classically simulating intermediate-scale instantaneous quantum polynomial circuits through a random graph approach

Hybrid quantum neural networks: harnessing dressed quantum circuits for enhanced tsunami prediction via earthquake data fusion

Tsunami is one of the deadliest natural disasters which can occur, leading to great loss of life and property. This study focuses on predicting tsunamis, using earthquake dataset from the year 1995 to 2023. The research introduces the Hybrid Quantum Neural Network (HQNN), an innovative model that combines Neural Network (NN) architecture with Parameterized Quantum Circuits (PmQC) to tackle complex machine learning (ML) problems where deep learning (DL) models struggle, aiming for higher accuracy in prediction while maintaining a compact model size. The hybrid model’s performance is compared with the classical model counterpart to investigate the quantum circuit’s effectivity as a layer in a DL model. The model has been implemented using 2-6 features through Principle Component Analysis (PCA) method. HQNN’s quantum circuit is a combination of Pennylane’s embedding (Angle Embedding (AE) and Instantaneous Quantum Polynomial (IQP) Embedding) and layer circuits (Basic Entangler Layers (BEL), Random Layers (RL), and Strongly Entangling Layers (SEL)), along with the classical layers. Results show that the proposed model achieved high performance, with a maximum accuracy up to 96.03% using 4 features with the combination of AE and SEL, superior to the DL model. Future research could explore the scalability and diverse applications of HQNN, as well as its potential to address practical ML challenges.

Existential unforgeability in quantum authentication from quantum physical unclonable functions based on random von Neumann measurement

Physical unclonable functions (PUFs) leverage inherent, nonclonable physical randomness to generate unique input-output pairs, serving as secure fingerprints for cryptographic protocols like authentication. Quantum PUFs (QPUFs) extend this concept by using quantum states as input-output pairs, offering advantages over classical PUFs, such as challenge reusability via public channels and eliminating the need for trusted parties due to the no-cloning theorem. Recent work introduced a generalized mathematical framework for QPUFs. It was shown that random unitary QPUFs cannot achieve existential unforgeability against quantum polynomial time (QPT) adversaries. Security was possible only with additional uniform randomness. To avoid the cost of external randomness, we propose a novel measurement-based scheme. Here, the randomness naturally arises from quantum measurements. Additionally, we introduce a second model where the QPUF functions as a nonunitary quantum channel, which guarantees existential unforgeability. These are the first models in the literature to demonstrate a high level of provable security. Finally, we show that the quantum phase estimation (QPE) protocol, applied to a Haar random unitary, serves as an approximate implementation of the second type of QPUF by approximating a von Neumann measurement on the unitary's eigenbasis. Published by the American Physical Society 2024

Efficient computations in central simple algebras using Amitsur cohomology

We introduce a presentation for central simple algebras over a field k using Amitsur cohomology. We provide efficient algorithms for computing a cocycle corresponding to any such algebra given by structure constants. If k is a number field, we use this presentation to prove that the explicit isomorphism problem (i.e., finding an isomorphism between central simple algebras given by structure constants) reduces to S-unit group computation and other related number theoretical computational problems. This also yields, conditionally on the generalised Riemann hypothesis, the first polynomial quantum algorithm for the explicit isomorphism problem over number fields.

On the quantum security of high-dimensional RSA protocol

Abstract The idea of extending the classical RSA protocol using algebraic number fields was introduced by Takagi and Naito (Construction of RSA cryptosystem over the algebraic field using ideal theory and investigation of its security. Electron Commun Japan Part III Fund Electr Sci. 2000;83:19–29). Recently, Zheng et al. proposed the use of the ring of algebraic integers of an algebraic number field and the lattice theory to present a high-dimensional form of RSA. The authors claim that their proposal is post-quantum and is significant both from the theoretical and practical point of view. In this article, we prove that the security of Zheng et al.’s scheme is still based on the factorization problem, and we present a practical quantum attack on this proposed scheme, our attack is a quantum polynomial time algorithm that employs Shor’s algorithm as a subroutine.

Polynomial quantum computing algorithms for solving the dualization problem for positive Boolean functions

Polynomial quantum computing algorithms for solving the dualization problem for positive Boolean functions

Quantum computational complexity and symmetry

Testing the symmetries of quantum states and channels provides a way to assess their usefulness for different physical, computational, and communication tasks. Here, we establish several complexity-theoretic results that classify the difficulty of symmetry-testing problems involving a unitary representation of a group and a state or a channel that is being tested. In particular, we prove that various such symmetry-testing problems are complete for bounded-error quantum polynomial time, quantum Merlin–Arthur (QMA), quantum statistical zero-knowledge, two-message quantum interactive proofs (QIPs), two-message QIPs restricted to entanglement-breaking provers, and QIPs, thus spanning the prominent classes of the QIP hierarchy and forging a nontrivial connection between symmetry and quantum computational complexity. Finally, we prove the inclusion of two Hamiltonian symmetry-testing problems in QMA and quantum Arthur–Merlin, while leaving it as an intriguing open question to determine whether these problems are complete for these classes.

Finding Zero Correlation Linear Hulls Using Quantum Bernstein-vazirani Algorithm

Abstract Facing the threat of quantum computing to cryptography schemes, it is urgent to study the applications of quantum algorithms to cryptanalysis and evaluate the security of cryptographic schemes in a quantum computing environment. To do this, it is necessary to analyze the ability of cryptanalysis methods, such as integral and linear attacks, when combined with quantum algorithms. In this article, we investigate the properties of block ciphers with SPN structure and their security against a quantum adversary and apply the Bernstein-Vazirani algorithm to zero correlation linear attacks. Based on the relation between linear approximates of SPN ciphers and the dual cipher’s linear structures, we proposed a quantum algorithm that can find linear hulls with zero correlation of SPN ciphers. We show the validity of the proposed quantum algorithm and estimate the corresponding computational complexity. Implementing our quantum algorithm only costs polynomial qubits and quantum gates.

Quantum Truncated Differential and Boomerang Attack

In order to design quantum-safe block ciphers, it is crucial to investigate the application of quantum algorithms to cryptographic analysis tools. In this study, we use the Bernstein–Vazirani algorithm to enhance truncated differential cryptanalysis and boomerang cryptanalysis. We first propose a quantum algorithm for finding truncated differentials, then rigorously prove that the output truncated differentials must have high differential probability for the vast majority of keys in the key space. Subsequently, based on this algorithm, we design a quantum algorithm for finding boomerang distinguishers. The quantum circuits of the two proposed quantum algorithms contain only polynomial quantum gates and qubits. Compared with classical tools for searching truncated differentials or boomerang distinguishers, the proposed algorithms can maintain the polynomial complexity while fully considering the impact of S-boxes and key scheduling.

Automorphisms of quantum polynomial rings and Drinfeld Hecke algebras

We consider quantum (skew) polynomial rings and observe that their graded automorphisms coincide with those of quantum exterior algebras. This allows us to define a quantum determinant that gives a homomorphism of groups acting on quantum polynomial rings. We use quantum subdeterminants to classify the resulting Drinfeld Hecke algebras for the symmetric group, other infinite families of Coxeter and complex reflection groups, and mystic reflection groups (which satisfy a version of the Shephard–Todd–Chevalley theorem). This direct combinatorial approach replaces the technology of Hochschild cohomology used by Naidu and Witherspoon over fields of characteristic zero and allows us to extend some of their results to fields of arbitrary characteristic and also locate new deformations of skew group algebras.

Universal control of a bosonic mode via drive-activated native cubic interactions

Linear bosonic modes offer a hardware-efficient alternative for quantum information processing but require access to some nonlinearity for universal control. The lack of nonlinearity in photonics has led to encoded measurement-based quantum computing, which relies on linear operations but requires access to resourceful (’nonlinear’) quantum states, such as cubic phase states. In contrast, superconducting microwave circuits offer engineerable nonlinearities but suffer from static Kerr nonlinearity. Here, we demonstrate universal control of a bosonic mode composed of a superconducting nonlinear asymmetric inductive element (SNAIL) resonator, enabled by native nonlinearities in the SNAIL element. We suppress static nonlinearities by operating the SNAIL in the vicinity of its Kerr-free point and dynamically activate nonlinearities up to third order by fast flux pulses. We experimentally realize a universal set of generalized squeezing operations, as well as the cubic phase gate, and exploit them to deterministically prepare a cubic phase state in 60 ns. Our results initiate the experimental field of polynomial quantum computing, in the continuous-variables notion originally introduced by Lloyd and Braunstein.

Exploring the neighborhood of 1-layer QAOA with instantaneous quantum polynomial circuits

We embed 1-layer QAOA circuits into the larger class of parametrized instantaneous quantum polynomial circuits to produce an improved variational quantum algorithm for solving combinatorial optimization problems. The use of analytic expressions to find optimal parameters classically makes our protocol robust against barren plateaus and hardware noise. The average overlap with the ground state scales as 2−0.31(2)N with the number of qubits N for random Sherrington-Kirkpatrick (SK) Hamiltonians of up to 29 qubits, a polynomial improvement over 1-layer QAOA. Additionally, we observe that performing variational imaginary time evolution on the manifold approximates low-temperature pseudo-Boltzmann states. Our protocol outperforms 1-layer QAOA on the recently released Quantinuum H2 trapped-ion quantum hardware and emulator, where we obtain an average approximation ratio of 0.985 across 312 random SK instances of 7 to 32 qubits, from which almost 44% are solved optimally using only 4 to 1208 shots per instance. Published by the American Physical Society 2024

On the complexity of quantum link prediction in complex networks

Link prediction methods use patterns in known network data to infer which connections may be missing. Previous work has shown that continuous-time quantum walks can be used to represent path-based link prediction, which we further study here to develop a more optimized quantum algorithm. Using a sampling framework for link prediction, we analyze the query access to the input network required to produce a certain number of prediction samples. Considering both well-known classical path-based algorithms using powers of the adjacency matrix as well as our proposed quantum algorithm for path-based link prediction, we argue that there is a polynomial quantum advantage on the dependence on N, the number of nodes in the network. We further argue that the complexity of our algorithm, although sub-linear in N, is limited by the complexity of performing a quantum simulation of the network’s adjacency matrix, which may prove to be an important problem in the development of quantum algorithms for network science in general.

The Challenges and Countermeasures of Ideological and Political Education Theory Course Teaching for Art Students in Universities in the New Media Era

Contemporary art students in universities are a generation that has grown up in the context of new media. With the increasingly close integration of art and new media, in order to enable art students to better shoulder the mission of spreading the excellent culture of the Chinese nation, ideological and political course teachers in universities should strengthen the reform and innovation of ideological and political theory courses for art students in teaching content, teaching methods, and the use of new media, Ensure that art students embody a positive energy of thought in their artistic creation. In this study, a crude oil price prediction model based on quantum leap radius and polynomial fitting is established by building a market atom model, fitting financial funds as "market atoms" with energy, and applying quantum leap theory to describe the leaping behavior of price. The LSTM neural network model is used to learn the price jump paths of financial instruments to improve the prediction effect. After 2000 rounds of iterative training, the RMSE of the model reaches 3.1136, and the correlation coefficient is 0.48084; the model's loss function decreases rapidly, indicating that the training effect is good. The study reveals the potential value of quantum theory in finance, preliminarily verifies the application of quantum leap and LSTM model in crude oil price prediction, and provides new ideas for the in-depth study of quantum finance theory.

An anharmonic alliance: exact WKB meets EPT

Certain quantum mechanical systems with a discrete spectrum, whose observables are given by a transseries in ħ, were shown to admit ħ0-deformations with Borel resummable expansions which reproduce the original model at ħ0 = ħ. Such expansions were dubbed Exact Perturbation Theory (EPT). We investigate how the above results can be obtained within the framework of the exact WKB method by studying the spectrum of polynomial quantum mechanical systems. Within exact WKB, energy eigenvalues are determined by exact quantization conditions defined in terms of Voros symbols aγi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {a}_{\\gamma_i} $$\\end{document}, γi being their associated cycles, and generally give rise to transseries in ħ. After reviewing how the Borel summability of energy eigenvalues in the quartic anharmonic potential emerges in exact WKB, we extend it to higher order anharmonic potentials with quantum corrections. We then show that any polynomial potential can be ħ0-deformed to a model where the exact quantization condition reads simply aγ = −1 and leads to the EPT Borel resummable series for all energy eigenvalues.

Quantum Obfuscation of Generalized Quantum Power Functions with Coefficient.

Quantum obfuscation is one of the important primitives in quantum cryptography that can be used to enhance the security of various quantum cryptographic schemes. The research on quantum obfuscation focuses mainly on the obfuscatability of quantum functions. As a primary quantum function, the quantum power function has led to the development of quantum obfuscation because it is applicable to construct new obfuscation applications such as quantum encryption schemes. However, the previous definition of quantum power functions is constrained and cannot be beneficial to the further construction of other quantum functions. Thus, it is essential to extend the definition of the basic quantum power function in a more general manner. In this paper, we provide a formal definition of two quantum power functions called generalized quantum power functions with coefficients, each of which is characterized by a leading coefficient and an exponent that corresponds to either a quantum or classical state, indicating the generality. The first is the quantum power function with a leading coefficient, and the second is the quantum n-th power function, which are both fundamental components of quantum polynomial functions. In addition, obfuscation schemes for the functions are constructed by quantum teleportation and quantum superdense coding, and demonstrations of their obfuscatability are also provided in this paper. This work establishes the fundamental basis for constructing more quantum functions that can be utilized for quantum obfuscation, therefore contributing to the theory of quantum obfuscation.

Mathematical modeling of resource allocation for cognitive radio sensor health monitoring system using coevolutionary quantum-behaved particle swarm optimization

Independent interactive medical applications cooperate closely with user systems through independent information and data exchange, which realizes digital and intelligent communication for health monitoring system. Due to extensive connectivity and application services, heavy data traffic and limited energy consumption reduce the quality of application services. In this paper, a mathematical model for efficient energy harvesting resource allocation based on cognitive radio sensor network (CRSN) infrastructure is constructed to address energy limitation issues in health monitoring system applications. Because of the highly dynamic nature of available network resources, energy efficiency and communication reliability can be improved. Since the proposed optimization problem is a complex and NP hard problem, coevolutionary theory is adopted, and then a stochastic optimization algorithm based on polynomial approximation and quantum particle swarm optimization is proposed. Simulation results show that the proposed CQPSO algorithm has better performance in terms of re-transmission probability, system throughput, and energy consumption compared to other similar algorithms. Efficient resource allocation improves the performance of the health monitoring system, more effectively and real-time monitoring of human physiological and motion information, providing testers with high-precision quantitative information, and achieving early monitoring and intervention of common diseases.

On the Need for Large Quantum Depth

Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “ Any quantum polynomial-time algorithm can be implemented with only O (log n ) quantum depth interspersed with polynomial-time classical computations. ” This can be formalized as asserting the equivalence of BQP and “ BQNC BPP .” However, Aaronson conjectured that “ there exists an oracle separation between BQP and BPP BQNC . ” BQNC BPP and BPP BQNC are two natural and seemingly incomparable ways of hybrid classical-quantum computation. In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d , there exists an oracle that separates quantum depth d and 2 d +1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.