Quantum Computational Speedup Research Articles

Uncertainties and coherence in DQC1

Correlations have been identified as a necessary ingredient for the speedup of certain mixed-state quantum computation. In this work, we analyze the model of deterministic quantum computation with one bit (DQC1) and study its information-theoretic aspects involving state-channel interaction. This state-channel interaction provides a new perspective for investigating DQC1. More specifically, by calculating some important information-theoretic quantities related to the state-channel interaction, such as uncertainties, coherence of quantum states as measured with respect to channels, etc., we reveal some basic features of quantum resources for the quantum speedup in this model. Moreover, since the state-channel framework is general enough to encompass all computational models, the studies performed here can be similarly applied to analyze other computational models from the perspective of quantum resources.

Quantum Logic under Semiclassical Limit: Information Loss

We consider the quantum computation efficiency from a new perspective. The efficiency is reduced to its classical counterpart by imposing the semiclassical limit. We show that this reduction is caused by the fact that any elementary quantum logic operation (gate) suffers the information loss during the transition to its classical analog. Amount of the information lost is estimated for any gate from the complete set. We demonstrate that the largest loss is obtained for non-commuting gates. This allows us to consider the non-commutativity as the quantum computational speed-up resource. Our method allows us to quantify advantages of a quantum computation as compared to the classical one by the direct analysis of the involved basic logic. The obtained results are illustrated by the application to a quantum discrete Fourier transform and Grover search algorithms.

Quantum diffusion map for nonlinear dimensionality reduction

Inspired by random walk on graphs, diffusion map (DM) is a class of unsupervised machine learning that offers automatic identification of low-dimensional data structure hidden in a high-dimensional dataset. In recent years, among its many applications, DM has been successfully applied to discover relevant order parameters in many-body systems, enabling automatic classification of quantum phases of matter. However, classical DM algorithm is computationally prohibitive for a large dataset, and any reduction of the time complexity would be desirable. With a quantum computational speedup in mind, we propose a quantum algorithm for DM, termed quantum diffusion map (qDM). Our qDM takes as an input $N$ classical data vectors, performs an eigen-decomposition of the Markov transition matrix in time $O(\log^3 N)$, and classically constructs the diffusion map via the readout (tomography) of the eigenvectors, giving a total expected runtime proportional to $N^2 \text{polylog}\, N$. Lastly, quantum subroutines in qDM for constructing a Markov transition matrix, and for analyzing its spectral properties can also be useful for other random walk-based algorithms.

Unobservable causal loops as a way to explain both the quantum computational speedup and quantum nonlocality

We consider the reversible processes between two one-to-one correlated measurement outcomes which underlie both problem solving and quantum nonlocality. In the former case, the two outcomes are the setting and the solution of the problem; in the latter, they are those of measuring a pair of maximally entangled observables whose subsystems are space separate. We argue that the quantum description of these processes mathematically describes the correlation but leaves the causal structure that physically ensures it free of violating the time symmetry required of the description of a reversible process. It would therefore be incomplete and could be completed by time symmetrizing it. This is done by assuming that the two measurements evenly contribute to selecting the pair of correlated measurement outcomes. Time symmetrization leaves the ordinary quantum description unaltered but shows that it is the quantum superposition of unobservable time-symmetrization instances whose causal structure is completely defined. Each instance is a causal loop: Causation goes from the initial to the final measurement outcome and then back from the final to the initial outcome. In the speedup, all is as if the problem solver knew in advance half of the information about the solution she will produce in the future and could use this knowledge to produce the solution with fewer computation steps. In nonlocality, the measurement on either subsystem retrocausally and locally changes the state of both subsystems when the two were not yet spatially separate. This locally causes the correlation between the two future measurement outcomes.

Closing Gaps of a Quantum Advantage with Short-Time Hamiltonian Dynamics.

Demonstrating a quantum computational speed-up is a crucial milestone for near-term quantum technology. Recently, sampling protocols for quantum simulators have been proposed that have the potential to show such a quantum advantage, based on commonly made assumptions. The key challenge in the theoretical analysis of this scheme-as of other comparable schemes such as boson sampling-is to lessen the assumptions and close the theoretical loopholes, replacing them by rigorous arguments. In this work, we prove two open conjectures for a simple sampling protocol that is based on the continuous time evolution of a translation-invariant Ising Hamiltonian: anticoncentration of the generated probability distributions and average-case hardness of exactly evaluating those probabilities. The latter is proven building upon recently developed techniques for random circuit sampling. For the former, we exploit the insight that approximate 2-designs for the unitary group admit anticoncentration. We then develop new techniques to prove that the 2D time evolution of the protocol gives rise to approximate 2-designs. Our work provides the strongest theoretical evidence to date that Hamiltonian quantum simulators are classically intractable.

Entanglement in Phase Estimation Algorithm and Quantum Counting Algorithm

For some certain problems, quantum algorithms are theoretically able to solve them quickly than classical algorithms. But the role of entanglement in achieving the quantum computational speedup is not fully understood. By theoretical analysis and numerical calculation of four practical use cases, we investigate the entanglement features of the quantum states employed in quantum phase estimation algorithm and quantum counting algorithm. The results show that whether these two algorithms generate entanglement depend on whether the input quantum state of the second register is a superposition state of the eigenstates.

A Relational Time-Symmetric Framework for Analyzing the Quantum Computational Speedup

The usual representation of quantum algorithms is limited to the process of solving the problem. We extend it to the process of setting the problem. Bob, the problem setter, selects a problem-setting by the initial measurement. Alice, the problem solver, unitarily computes the corresponding solution and reads it by the final measurement. This simple extension creates a new perspective from which to see the quantum algorithm. First, it highlights the relevance of time-symmetric quantum mechanics to quantum computation: the problem-setting and problem solution, in their quantum version, constitute pre- and post-selection, hence the process as a whole is bound to be affected by both boundary conditions. Second, it forces us to enter into relational quantum mechanics. There must be a representation of the quantum algorithm with respect to Bob, and another one with respect to Alice, from whom the outcome of the initial measurement, specifying the setting and thus the solution of the problem, must be concealed. Time-symmetrizing the quantum algorithm to take into account both boundary conditions leaves the representation to Bob unaltered. It shows that the representation to Alice is a sum over histories in each of which she remains shielded from the information coming to her from the initial measurement, not from that coming to her backwards in time from the final measurement. In retrospect, all is as if she knew in advance, before performing her problem-solving action, half of the information that specifies the solution of the problem she will read in the future and could use this information to reach the solution with fewer computation steps (oracle queries). This elucidates the quantum computational speedup in all the quantum algorithms examined.

Massively Parallel Coincidence Counting of High-Dimensional Entangled States

Entangled states of light are essential for quantum technologies and fundamental tests of physics. Current systems rely on entanglement in 2D degrees of freedom, e.g., polarization states. Increasing the dimensionality provides exponential speed-up of quantum computation, enhances the channel capacity and security of quantum communication protocols, and enables quantum imaging; unfortunately, characterizing high-dimensional entanglement of even bipartite quantum states remains prohibitively time-consuming. Here, we develop and experimentally demonstrate a new theory of camera detection that leverages the massive parallelization inherent in an array of pixels. We show that a megapixel array, for example, can measure a joint Hilbert space of 1012 dimensions, with a speed-up of nearly four orders-of-magnitude over traditional methods. The technique uses standard geometry with existing technology, thus removing barriers of entry to quantum imaging experiments, generalizes readily to arbitrary numbers of entangled photons, and opens previously inaccessible regimes of high-dimensional quantum optics.

Quantum superpositions cannot be epistemic

Quantum superposition states are behind many of the curious phenomena exhibited by quantum systems, including Bell non-locality, quantum interference, quantum computational speed-up, and the measurement problem. At the same time, many qualitative properties of quantum superpositions can also be observed in classical probability distributions leading to a suspicion that superpositions may be explicable as probability distributions over less problematic states, that is, a suspicion that superpositions are \emph{epistemic}. Here, it is proved that, for any quantum system of dimension $d>3$, this cannot be the case for almost all superpositions. Equivalently, any underlying ontology must contain ontic superposition states. A related question concerns the more general possibility that some pairs of non-orthogonal quantum states $|\psi\rangle,|\phi\rangle$ could be ontologically indistinct (there are ontological states which fail to distinguish between these quantum states). A similar method proves that if $|\langle\phi|\psi\rangle|^{2}\in(0,\frac{1}{4})$ then $|\psi\rangle,|\phi\rangle$ must approach ontological distinctness as $d\rightarrow\infty$. The robustness of these results to small experimental error is also discussed.

Necessity of Superposition of Macroscopically Distinct States for Quantum Computational Speedup

For quantum computation, we investigate the conjecture that the superposition of macroscopically distinct states is necessary for a large quantum speedup. Although this conjecture was supported for a circuit-based quantum computer performing Shor's factoring algorithm [A. Ukena and A. Shimizu: Phys. Rev. A 69 (2004) 022301], it needs to be generalized for it to be applicable to a large class of algorithms and/or other models such as measurement-based quantum computers. To treat such general cases, we first generalize the indices for the superposition of macroscopically distinct states. We then generalize the conjecture, using the generalized indices, in such a way that it is unambiguously applicable to general models if a quantum algorithm achieves exponential speedup. On the basis of this generalized conjecture, we further extend the conjecture to Grover's quantum search algorithm, whose speedup is large but quadratic. It is shown that this extended conjecture is also correct. Since Grover's algorithm is a representative algorithm for unstructured problems, the present result further supports the conjecture.

Negative quasi-probability as a resource for quantum computation

A central problem in quantum information is to determine the minimal physical resources that are required for quantum computational speed-up and, in particular, for fault-tolerant quantum computation. We establish a remarkable connection between the potential for quantum speed-up and the onset of negative values in a distinguished quasi-probability representation, a discrete analogue of the Wigner function for quantum systems of odd dimension. This connection allows us to resolve an open question on the existence of bound states for magic state distillation: we prove that there exist mixed states outside the convex hull of stabilizer states that cannot be distilled to non-stabilizer target states using stabilizer operations. We also provide an efficient simulation protocol for Clifford circuits that extends to a large class of mixed states, including bound universal states.

Entanglement of periodic states, the quantum Fourier transform, and Shor’s factoring algorithm

The preprocessing stage of Shor's algorithm generates a class of quantum states referred to as periodic states, on which the quantum Fourier transform is applied. Such states also play an important role in other quantum algorithms that rely on the quantum Fourier transform. Since entanglement is believed to be a necessary resource for quantum computational speedup, we analyze the entanglement of periodic states and the way it is affected by the quantum Fourier transform. To this end, we derive a formula that evaluates the Groverian entanglement measure for periodic states. Using this formula, we explain the surprising result that the Groverian entanglement of the periodic states built up during the preprocessing stage is only slightly affected by the quantum Fourier transform.

Quantum stochasticity and neuronal computations

AbstractThe nervous system probably cannot display macroscopic quantum (i.e. classically impossible) behaviours such as quantum entanglement, superposition or tunnelling (Koch and Hepp, Nature 440:611, 2006). However, in contrast to this quantum ‘mysticism’ there is an alternative way in which quantum events might influence the brain activity. The nervous system is a nonlinear system with many feedback loops at every level of its structural hierarchy. A conventional wisdom is that in macroscopic objects the quantum fluctuations are self-averaging and thus not important. Nevertheless this intuition might be misleading in the case of nonlinear complex systems. Because of a high sensitivity to initial conditions, in chaotic systems the microscopic fluctuations may be amplified upward and thereby affect the system's output. In this way stochastic quantum dynamics might sometimes alter the outcome of neuronal computations, not by generating classically impossible solutions, but by influencing the selection of many possible solutions (Satinover, Quantum Brain, Wiley & Sons, 2001). I am going to discuss recent theoretical proposals and experimental findings in quantum mechanics, complexity theory and computational neuroscience suggesting that biological evolution is able to take advantage of quantum-computational speed-up. I predict that the future research on quantum complex systems will provide us with novel interesting insights that might be relevant also for neurobiology and neurophilosophy.

Most Quantum States Are Too Entangled To Be Useful As Computational Resources

It is often argued that entanglement is at the root of the speedup for quantum compared to classical computation, and that one needs a sufficient amount of entanglement for this speedup to be manifest. In measurement-based quantum computing, the need for a highly entangled initial state is particularly obvious. Defying this intuition, we show that quantum states can be too entangled to be useful for the purpose of computation, in that high values of the geometric measure of entanglement preclude states from offering a universal quantum computational speedup. We prove that this phenomenon occurs for a dramatic majority of all states: the fraction of useful n-qubit pure states is less than exp(-n;{2}). This work highlights a new aspect of the role entanglement plays for quantum computational speedups.

Role of entanglement and correlations in mixed-state quantum computation

In a quantum computation with pure states, the generation of large amounts of entanglement is known to be necessary for a speedup with respect to classical computations. However, examples of quantum computations with mixed states are known, such as the deterministic computation with one quantum qubit (DQC1) model [Knill and Laflamme, Phys. Rev. Lett. 81, 5672 (1998)], in which entanglement is at most marginally present, and yet a computational speedup is believed to occur. Correlations, and not entanglement, have been identified as a necessary ingredient for mixed-state quantum computation speedups. Here we show that correlations, as measured through the operator Schmidt rank, are indeed present in large amounts in the DQC1 circuit. This provides evidence for the preclusion of efficient classical simulation of DQC1 by means of a whole class of classical simulation algorithms, thereby reinforcing the conjecture that DQC1 leads to a genuine quantum computational speedup.

Discrete Wigner functions and quantum computational speedup

Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] have recently defined a class of discrete Wigner functions $W$ to represent quantum states in a finite Hilbert space dimension $d$. I characterize the set ${C}_{d}$ of states having non-negative $W$ simultaneously in all definitions of $W$ in this class. For $d\ensuremath{\leqslant}5$ I show ${C}_{d}$ is the convex hull of stabilizer states. This supports the conjecture that negativity of $W$ is necessary for exponential speedup in pure-state quantum computation.

On the role of entanglement in quantum-computational speed-up

For any quantum algorithm operating on pure states we prove that the presence of multi-partite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the quantum algorithm is to offer an exponential speed-up over classical computation. Furthermore we prove that the algorithm can be classically efficiently simulated to within a prescribed tolerance \eta even if a suitably small amount of global entanglement (depending on \eta) is present. We explicitly identify the occurrence of increasing multi-partite entanglement in Shor's algorithm. Our results do not apply to quantum algorithms operating on mixed states in general and we discuss the suggestion that an exponential computational speed-up might be possible with mixed states in the total absence of entanglement. Finally, despite the essential role of entanglement for pure state algorithms, we argue that it is nevertheless misleading to view entanglement as a key resource for quantum computational power.