Catalytic space: Non-determinism and hierarchy

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation

Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks

Many computational problems are subject to a quantum speed-up: one might find that a problem having an Opn3q-time or Opn2q-time classic algorithm can be solved by a known Opn1.5q-time or Opnq-time quantum algorithm. The question naturally arises: how much quantum speed-up is possible? The area of fine-grained complexity allows us to prove optimal lower-bounds on the complexity of various computational problems, based on the conjectured hardness of certain natural, well-studied problems. This theory has recently been extended to the quantum setting, in two independent papers by Buhrman, Patro and Speelman [7], and by Aaronson, Chia, Lin, Wang, and Zhang [1]. In this paper, we further extend the theory of fine-grained complexity to the quantum setting. A fundamental conjecture in the classical setting states that the 3SUM problem cannot be solved by (classical) algorithms in time Opn2´εq, for any ε ą 0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear Opn1´εq-time quantum algorithms for the 3SUM problem. Based on the Quantum-3SUM-Conjecture, we show new lower-bounds on the time complexity of quantum algorithms for several computational problems. Most of our lower-bounds are optimal, in that they match known upper-bounds, and hence they imply tight limits on the quantum speedup that is possible for these problems. These results are proven by adapting to the quantum setting known classical fine-grained reductions from the 3SUM problem. This adaptation is not trivial, however, since the original classical reductions require pre-processing the input in various ways, e.g. by sorting it according to some order, and this pre-processing (provably) cannot be done in sublinear quantum time. We overcome this bottleneck by combining a quantum walk with a classical dynamic data-structure having a certain “history-independence” property. This type of construction has been used in the past to prove upper bounds, and here we use it for the first time as part of a reduction. This general proof strategy allows us to prove tight lower bounds on several computational-geometry problems, on Convolution-3SUM and on the 0-Edge-Weight-Triangle problem, conditional on the Quantum-3SUM-Conjecture. We believe this proof strategy will be useful in proving tight (conditional) lower-bounds, and limits on quantum speed-ups, for many other problems

Catalytic space: non-determinism and hierarchy

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation

Asymptotic performance of port-based teleportation

Quantum teleportation is one of the fundamental building blocks of quantum Shannon theory. While ordinary teleportation is simple and efficient, port-based teleportation (PBT) enables applications such as universal programmable quantum processors, instantaneous non-local quantum computation and attacks on position-based quantum cryptography. In this work, we determine the fundamental limit on the performance of PBT: for arbitrary fixed input dimension and a large number N of ports, the error of the optimal protocol is proportional to the inverse square of N. We prove this by deriving an achievability bound, obtained by relating the corresponding optimization problem to the lowest Dirichlet eigenvalue of the Laplacian on the ordered simplex. We also give an improved converse bound of matching order in the number of ports. In addition, we determine the leading-order asymptotics of PBT variants defined in terms of maximally entangled resource states. The proofs of these results rely on connecting recently-derived representation-theoretic formulas to random matrix theory. Along the way, we refine a convergence result for the fluctuations of the Schur-Weyl distribution by Johansson, which might be of independent interest

Distribution and coexistence of myoclonus and dystonia as clinical predictors of SGCE mutation status: a pilot study

Introduction: Myoclonus-dystonia (M-D) is a young onset movement disorder typically involving myoclonus and dystonia of the upper body. A proportion of the cases are caused by mutations to the autosomal dominantly inherited, maternally imprinted, epsilon-sarcoglycan gene (SGCE). Despite several sets of diagnostic criteria, identification of patients most likely to have an SGCE mutation remains difficult. Methods: Forty consecutive patients meeting pre-existing diagnostic clinical criteria for M-D underwent a standardized clinical examination (20 SGCE mutation positive and 20 negative). Each video was reviewed and systematically scored by two assessors blinded to mutation status. In addition, the presence and coexistence of myoclonus and dystonia was recorded in four body regions (neck, arms, legs, and trunk) at rest and with action. Results: Thirty-nine patients were included in the study (one case was excluded owing to insufficient video footage). Based on previously proposed diagnostic criteria, patients were subdivided into 24 "definite," 5 "probable," and 10 "possible" M-D. Motor symptom severity was higher in the SGCE mutation-negative group. Myoclonus and dystonia were most commonly observed in the neck and upper limbs of both groups. Truncal dystonia with action was significantly seen more in the mutation-negative group (p <0.05). Coexistence of myoclonus and dystonia in the same body part with action was more commonly seen in the mutation-negative cohort (p <0.05). Conclusion: Truncal action dystonia and coexistence of myoclonus and dystonia in the same body part with action might suggest the presence of an alternative mutation in patients with M-D

Deep Brain Stimulation of the Pallidum is Effective and Might Stabilize Striatal D2 Receptor Binding in Myoclonus–Dystonia

Purpose: To assess clinical efficacy of deep brain stimulation (DBS) of the pallidum in Myoclonus–Dystonia (M–D) patients, and to compare pre- and post-operative striatal dopamine D2 receptor availability. Methods: Clinical parameters were scored using validated rating scales for myoclonus and dystonia. Dopamine D2 receptor binding of three patients was studied before surgery and approximately 2 years post-operatively using 123-I-iodobenzamide Single Photon Emission Computed Tomography. Two patients who did not undergo surgery served as controls. Results: Clinically, the three M–D patients improved 83, 17, and 100%, respectively on the myoclonus rating scale and 78, 23, and 65% on the dystonia rating scale after DBS. Dopamine D2 receptor binding did not change after surgery. In the two control subjects, binding has lowered further. Conclusion: These findings confirm that DBS of the pallidum has beneficial effects on motor symptoms in M–D and suggest this procedure might stabilize dopamine D2 receptor binding