arXiv subjects

Dax Enshan Koh

Publications and source records attributed to Dax Enshan Koh.

At least 37 records · Page 2Linked to original sources

arXiv2025

Pseudorandom density matrices

Pseudorandom states (PRSs) are state ensembles that cannot be efficiently distinguished from Haar random states. However, the definition of PRSs has been limited to pure states and lacks robustness against noise. Here, we introduce pseudorandom density matrices (PRDMs), ensembles of $n$-qubit states that are computationally indistinguishable from the generalized Hilbert-Schmidt ensemble (GHSE), which is constructed from $(n+m)$-qubit Haar random states with $m$ qubits traced out. For $m=0$, PRDMs are equivalent to PRSs, whereas for $m=ω(\log n)$, PRDMs are computationally indistinguishable from the maximally mixed state. PRDMs with $m=ω(\log n)$ are robust to unital noise channels and separated in terms of security from PRS. PRDMs disguise valuable quantum resources, possessing near-maximal entanglement, magic and coherence, while being computationally indistinguishable from resource-free states. PRDMs exhibit a pseudoresource gap of $Θ(n)$ vs $0$, surpassing previously found gaps. We also render EFI pairs, a fundamental cryptographic primitive, robust to strong mixed unitary noise. Our work has major implications on quantum resource theory: We show that entanglement, magic and coherence cannot be efficiently tested, and that black-box resource distillation requires a superpolynomial number of copies. We also establish lower bounds on the purity needed for efficient testing and black-box distillation. Finally, we introduce memoryless PRSs, a noise-robust notion of PRS which are indistinguishable to Haar random states for efficient algorithms without quantum memory, as well as noise-robust quantum money. Our work provides a comprehensive framework of pseudorandomness for mixed states, which yields powerful quantum cryptographic primitives and fundamental bounds on quantum resource theories.

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arXiv2025

Pseudorandom quantum authentication

We introduce the pseudorandom quantum authentication scheme (PQAS), an efficient method for encrypting quantum states that relies solely on the existence of pseudorandom unitaries (PRUs). The scheme guarantees that for any eavesdropper with quantum polynomial-time (QPT) computational power, the encrypted states are indistinguishable from the maximally mixed state. Furthermore, the receiver can verify that the state has not been tampered with and recover the original state with asymptotically unit fidelity. Our scheme is cost-effective, requiring only polylogarithmic circuit depth and a single shared key to encrypt a polynomial number of states. Notably, the PQAS can potentially exist even without quantum-secure one-way functions, requiring fundamentally weaker computational assumptions than semantic classical cryptography. Additionally, PQAS is secure against attacks that plague protocols based on QPT indistinguishability from Haar random states, such as chosen-plaintext attacks (CPAs) and attacks that reveal meta-information such as quantum resources. We relate the amount of meta-information that is leaked to quantum pseudoresources, giving the concept a practical meaning. As an application, we construct important cryptographic primitives, such as verifiable pseudorandom density matrices (VPRDMs), which are QPT-indistinguishable from random mixed states while being efficiently verifiable via a secret key, as well as verifiable noise-robust EFI pairs and one-way state generators (OWSGs). Our results establish a new paradigm of quantum information processing with weaker computational assumptions.

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arXiv2024

Classical Shadows with Improved Median-of-Means Estimation

The classical shadows protocol, introduced by Huang et al. [Nat. Phys. 16, 1050 (2020)], makes use of the median-of-means (MoM) estimator to efficiently estimate the expectation values of $M$ observables with failure probability $δ$ using only $\mathcal{O}(\log(M/δ))$ measurements. In their analysis, Huang et al. used loose constants in their asymptotic performance bounds for simplicity. However, the specific values of these constants can significantly affect the number of shots used in practical implementations. To address this, we studied a modified MoM estimator proposed by Minsker [PMLR 195, 5925 (2023)] that uses optimal constants and involves a U-statistic over the data set. For efficient estimation, we implemented two types of incomplete U-statistics estimators, the first based on random sampling and the second based on cyclically permuted sampling. We compared the performance of the original and modified estimators when used with the classical shadows protocol with single-qubit Clifford unitaries (Pauli measurements) for an Ising spin chain, and global Clifford unitaries (Clifford measurements) for the Greenberger-Horne-Zeilinger (GHZ) state. While the original estimator outperformed the modified estimators for Pauli measurements, the modified estimators showed improved performance over the original estimator for Clifford measurements. Our findings highlight the importance of tailoring estimators to specific measurement settings to optimize the performance of the classical shadows protocol in practical applications.

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arXiv2024

Analytical Expressions for the Quantum Approximate Optimization Algorithm and its Variants

The quantum approximate optimization algorithm (QAOA) is a near-term quantum algorithm aimed at solving combinatorial optimization problems. Since its introduction, various generalizations have emerged, spanning modifications to the initial state, phase unitaries, and mixer unitaries. In this work, we present an analytical study of broad families of QAOA variants. We begin by examining a family of QAOA with product mixers, which includes single-body mixers parametrized by multiple variational angles, and derive exact analytical expressions for the cost expectation on weighted problem graphs in the single-layer ansatz setting. We then analyze a family of QAOA that employs many-body Grover-type mixers, deriving analogous analytical expressions for weighted problem hypergraphs in the setting of arbitrarily many circuit ansatz layers. For both families, we allow individual phase angles for each node and edge (hyperedge) in the problem graph (hypergraph). Our results reveal that, in contrast to product mixers, the Grover mixer is sensitive to contributions from cycles of all lengths in the problem graph, exhibiting a form of non-locality. Our study advances the understanding of QAOA's behavior in general scenarios, providing a foundation for further theoretical exploration.

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arXiv2024

Quantum Volunteer's Dilemma

The volunteer's dilemma is a well-known game in game theory that models the conflict players face when deciding whether to volunteer for a collective benefit, knowing that volunteering incurs a personal cost. In this work, we introduce a quantum variant of the classical volunteer's dilemma, generalizing it by allowing players to utilize quantum strategies. Employing the Eisert-Wilkens-Lewenstein quantization framework, we analyze a multiplayer quantum volunteer's dilemma scenario with an arbitrary number of players, where the cost of volunteering is shared equally among the volunteers. We derive analytical expressions for the players' expected payoffs and demonstrate the quantum game's advantage over the classical game. In particular, we prove that the quantum volunteer's dilemma possesses symmetric Nash equilibria with larger expected payoffs compared to the unique symmetric Nash equilibrium of the classical game, wherein players use mixed strategies. Furthermore, we show that the quantum Nash equilibria we identify are Pareto optimal. Our findings reveal distinct dynamics in volunteer's dilemma scenarios when players adhere to quantum rules, underscoring a strategic advantage of decision-making in quantum settings.

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arXiv2024

Teleportation with Embezzling Catalysts

Quantum teleportation is the process of transferring quantum information using classical communication and pre-shared entanglement. This process can benefit from the use of catalysts, which are ancillary entangled states that can enhance teleportation without being consumed. While chemical catalysts undergoing deactivation invariably exhibit inferior performance compared to those unaffected by deactivation, quantum catalysts, termed embezzling catalysts, that are subject to deactivation, may surprisingly outperform their non-deactivating counterparts. In this work, we present teleportation protocols with embezzling catalyst that can achieve arbitrarily high fidelity, namely the teleported state can be made arbitrarily close to the original state, with finite-dimensional embezzling catalysts. We show that some embezzling catalysts are universal, meaning that they can improve the teleportation fidelity for any pre-shared entanglement. We also explore methods to reduce the dimension of catalysts without increasing catalyst consumption, an essential step towards realizing quantum catalysis in practice.

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arXiv2024

Communication with Quantum Catalysts

Communication is essential for advancing science and technology. Quantum communication, in particular, benefits from the use of catalysts. During the communication process, these catalysts enhance performance while remaining unchanged. Although chemical catalysts that undergo deactivation typically perform worse than those that remain unaffected, quantum catalysts, referred to as embezzling catalysts, can surprisingly outperform their non-deactivating counterparts despite experiencing slight alterations. In this work, we employ embezzling quantum catalysts to enhance the transmission of both quantum and classical information. Our results reveal that using embezzling catalysts augments the efficiency of information transmission across noisy quantum channels, ensuring a non-zero catalytic channel capacity. Furthermore, we introduce catalytic superdense coding, demonstrating how embezzling catalysts can enhance the transmission of classical information. Finally, we explore methods to reduce the dimensionality of catalysts, a step toward making quantum catalysis a practical reality.

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arXiv2024

Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach

We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap history states. The proposed algorithm is not only efficient in time complexity, but has lower memory costs compared to classical methods. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation cost scales economically with the number of time steps. As a proof of concept, we apply our algorithm to solve a range of fractional partial differential equations commonly encountered in engineering applications, such as the sub-diffusion equation, the non-linear Burgers' equation and a coupled diffusive epidemic model. We assess quantum hardware performance under realistic noise conditions, further validating the practical utility of our algorithm.

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arXiv2024

Towards enhancing quantum expectation estimation of matrices through partial Pauli decomposition techniques and post-processing

We introduce an approach for estimating the expectation values of arbitrary $n$-qubit matrices $M \in \mathbb{C}^{2^n\times 2^n}$ on a quantum computer. In contrast to conventional methods like the Pauli decomposition that utilize $4^n$ distinct quantum circuits for this task, our technique employs at most $2^n$ unique circuits, with even fewer required for matrices with limited bandwidth. Termed the \textit{partial Pauli decomposition}, our method involves observables formed as the Kronecker product of a single-qubit Pauli operator and orthogonal projections onto the computational basis. By measuring each such observable, one can simultaneously glean information about $2^n$ distinct entries of $M$ through post-processing of the measurement counts. This reduction in quantum resources is especially crucial in the current noisy intermediate-scale quantum era, offering the potential to accelerate quantum algorithms that rely heavily on expectation estimation, such as the variational quantum eigensolver and the quantum approximate optimization algorithm.

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arXiv2024

Error-mitigated fermionic classical shadows on noisy quantum devices

Efficiently estimating fermionic Hamiltonian expectation values is vital for simulating various physical systems. Classical shadow (CS) algorithms offer a solution by reducing the number of quantum state copies needed, but noise in quantum devices poses challenges. We propose an error-mitigated CS algorithm assuming gate-independent, time-stationary, and Markovian (GTM) noise. For $n$-qubit systems, our algorithm, which employs the easily prepared initial state $|0^n\rangle\!\langle 0^n|$ assumed to be noiseless, efficiently estimates $k$-RDMs with $\widetilde{\mathcal O}(kn^k)$ state copies and $\widetilde{\mathcal O}(\sqrt{n})$ calibration measurements for GTM noise with constant fidelities. We show that our algorithm is robust against noise types like depolarizing, damping, and $X$-rotation noise with constant strengths, showing scalings akin to prior CS algorithms for fermions but with better noise resilience. Numerical simulations confirm our algorithm's efficacy in noisy settings, suggesting its viability for near-term quantum devices.

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arXiv2024

An Expressive Ansatz for Low-Depth Quantum Approximate Optimisation

The quantum approximate optimisation algorithm (QAOA) is a hybrid quantum-classical algorithm used to approximately solve combinatorial optimisation problems. It involves multiple iterations of a parameterised ansatz comprising a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on NISQ devices, physical limitations can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), an overparameterised variant of QAOA that assigns more classical parameters to the ansatz to improve its performance at low depths. XQAOA also introduces an additional Pauli-Y component in the mixer Hamiltonian, allowing the mixer to implement arbitrary unitary transformations on each qubit. To benchmark the performance of XQAOA at unit depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, Multi-Angle QAOA (MA-QAOA), a classical-relaxed algorithm, and the state-of-the-art Goemans-Williamson algorithm on a set of unweighted regular graphs with 128 and 256 nodes for degrees ranging from 3 to 10. Our results indicate that at unit depth, XQAOA has benign loss landscapes, allowing it to consistently outperform QAOA, MA-QAOA, and the classical-relaxed algorithm on all graph instances and the Goemans-Williamson algorithm on graph instances with degrees greater than 4. Small-scale simulations also reveal that unit-depth XQAOA surpasses both QAOA and MA-QAOA on all tested depths up to five. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and analytically show that for some graphs in this family, XQAOA achieves a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for variational quantum optimisation on NISQ devices, offering competitive performance at low depths.

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arXiv2023

A quantum tug of war between randomness and symmetries on homogeneous spaces

We explore the interplay between symmetry and randomness in quantum information. Adopting a geometric approach, we consider states as $H$-equivalent if related by a symmetry transformation characterized by the group $H$. We then introduce the Haar measure on the homogeneous space $\mathbb{U}/H$, characterizing true randomness for $H$-equivalent systems. While this mathematical machinery is well-studied by mathematicians, it has seen limited application in quantum information: we believe our work to be the first instance of utilizing homogeneous spaces to characterize symmetry in quantum information. This is followed by a discussion of approximations of true randomness, commencing with $t$-wise independent approximations and defining $t$-designs on $\mathbb{U}/H$ and $H$-equivalent states. Transitioning further, we explore pseudorandomness, defining pseudorandom unitaries and states within homogeneous spaces. Finally, as a practical demonstration of our findings, we study the expressibility of quantum machine learning ansatze in homogeneous spaces. Our work provides a fresh perspective on the relationship between randomness and symmetry in the quantum world.

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arXiv2023

Variational Quantum Simulation of Partial Differential Equations: Applications in Colloidal Transport

We assess the use of variational quantum imaginary time evolution for solving partial differential equations. Our results demonstrate that real-amplitude ansaetze with full circular entangling layers lead to higher-fidelity solutions compared to those with partial or linear entangling layers. To efficiently encode impulse functions, we propose a graphical mapping technique for quantum states that often requires only a single bit-flip of a parametric gate. As a proof of concept, we simulate colloidal deposition on a planar wall by solving the Smoluchowski equation including the Derjaguin-Landau-Verwey-Overbeek (DLVO) potential energy. We find that over-parameterization is necessary to satisfy certain boundary conditions and that higher-order time-stepping can effectively reduce norm errors. Together, our work highlights the potential of variational quantum simulation for solving partial differential equations using near-term quantum devices.

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arXiv2023

Fundamental Limitations on Communication over a Quantum Network

Entanglement, a fundamental feature of quantum mechanics, has long been recognized as a valuable resource in enabling secure communications and surpassing classical limits. However, previous research has primarily concentrated on static entangled states generated at a single point in time, overlooking the crucial role of the quantum dynamics responsible for creating such states. Here, we propose a framework for investigating entanglement across multiple time points, termed temporal entanglement, and demonstrate that the performance of a quantum network in transmitting information is inherently dependent on its temporal entanglement. Through case studies, we showcase the capabilities of our framework in enhancing conventional quantum teleportation and achieving exponential performance growth in the protocol of quantum repeaters. Additionally, our framework effectively doubles the communication distance in certain noise models. Our results address the longstanding question surrounding temporal entanglement within non-Markovian processes and its impact on quantum communication, thereby pushing the frontiers of quantum information science.

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arXiv2022

Wasserstein Complexity of Quantum Circuits

Given a unitary transformation, what is the size of the smallest quantum circuit that implements it? This quantity, known as the quantum circuit complexity, is a fundamental property of quantum evolutions that has widespread applications in many fields, including quantum computation, quantum field theory, and black hole physics. In this letter, we obtain a new lower bound for the quantum circuit complexity in terms of a novel complexity measure that we propose for quantum circuits, which we call the quantum Wasserstein complexity. Our proposed measure is based on the quantum Wasserstein distance of order one (also called the quantum earth mover's distance), a metric on the space of quantum states. We also prove several fundamental and important properties of our new complexity measure, which stand to be of independent interest. Finally, we show that our new measure also provides a lower bound for the experimental cost of implementing quantum circuits, which implies a quantum limit on converting quantum resources to computational resources. Our results provide novel applications of the quantum Wasserstein distance and pave the way for a deeper understanding of the resources needed to implement a quantum computation.

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arXiv2022

Connecting geometry and performance of two-qubit parameterized quantum circuits

Parameterized quantum circuits (PQCs) are a central component of many variational quantum algorithms, yet there is a lack of understanding of how their parameterization impacts algorithm performance. We initiate this discussion by using principal bundles to geometrically characterize two-qubit PQCs. On the base manifold, we use the Mannoury-Fubini-Study metric to find a simple equation relating the Ricci scalar (geometry) and concurrence (entanglement). By calculating the Ricci scalar during a variational quantum eigensolver (VQE) optimization process, this offers us a new perspective to how and why Quantum Natural Gradient outperforms the standard gradient descent. We argue that the key to the Quantum Natural Gradient's superior performance is its ability to find regions of high negative curvature early in the optimization process. These regions of high negative curvature appear to be important in accelerating the optimization process.

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arXiv2022

Classical Shadows With Noise

The classical shadows protocol, recently introduced by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], is a quantum-classical protocol to estimate properties of an unknown quantum state. Unlike full quantum state tomography, the protocol can be implemented on near-term quantum hardware and requires few quantum measurements to make many predictions with a high success probability. In this paper, we study the effects of noise on the classical shadows protocol. In particular, we consider the scenario in which the quantum circuits involved in the protocol are subject to various known noise channels and derive an analytical upper bound for the sample complexity in terms of a shadow seminorm for both local and global noise. Additionally, by modifying the classical post-processing step of the noiseless protocol, we define a new estimator that remains unbiased in the presence of noise. As applications, we show that our results can be used to prove rigorous sample complexity upper bounds in the cases of depolarizing noise and amplitude damping.

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arXiv2022

Exploring variational quantum eigensolver ansatzes for the long-range XY model

Finding the ground state energy and wavefunction of a quantum many-body system is a key problem in quantum physics and chemistry. We study this problem for the long-range XY model by using the variational quantum eigensolver (VQE) algorithm. We consider VQE ansatzes with full and linear entanglement structures consisting of different building gates: the CNOT gate, the controlled-rotation (CRX) gate, and the two-qubit rotation (TQR) gate. We find that the full-entanglement CRX and TQR ansatzes can sufficiently describe the ground state energy of the long-range XY model. In contrast, only the full-entanglement TQR ansatz can represent the ground state wavefunction with a fidelity close to one. In addition, we find that instead of using full-entanglement ansatzes, restricted-entanglement ansatzes where entangling gates are applied only between qubits that are a fixed distance from each other already suffice to give acceptable solutions. Using the entanglement entropy to characterize the expressive powers of the VQE ansatzes, we show that the full-entanglement TQR ansatz has the highest expressive power among them.

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