scholarly journals Composability of global phase invariant distance and its application to approximation error management

2021 ◽  
Author(s):  
Priyanka Mukhopadhyay
Keyword(s):  
Fault Tolerant ◽  
Quantum Algorithms ◽  
Approximation Error ◽  
Optimal Number ◽  
Operator Norm ◽  
Quantum Circuits ◽  
Frobenius Norm ◽  
Invariant Distance ◽  
Composition Product ◽  
Better Than

Abstract Many quantum algorithms can be written as a composition of unitaries, some of which can be exactly synthesized by a universal fault-tolerant gate set like Clifford+T, while others can be approximately synthesized. A quantum compiler synthesizes each approximately synthesizable unitary up to some approximation error, such that the error of the overall unitary remains bounded by a certain amount. In this paper we consider the case when the errors are measured in the global phase invariant distance. Apart from deriving a relation between this distance and the Frobenius norm, we show that this distance composes. If a unitary is written as a composition (product and tensor product) of other unitaries, we derive bounds on the error of the overall unitary as a function of the errors of the composed unitaries. Our bound is better than the sum-of-error bound, derived by Bernstein- Vazirani(1997), for the operator norm. This builds the intuition that working with the global phase invariant distance might give us a lower resource count while synthesizing quantum circuits. Next we consider the following problem. Suppose we are given a decomposition of a unitary, that is, the unitary is expressed as a composition of other unitaries. We want to distribute the errors in each component such that the resource-count (specifically T-count) is optimized. We consider the specific case when the unitary can be decomposed such that the $R_z(\theta)$ gates are the only approximately synthesizable component. We prove analytically that for both the operator norm and global phase invariant distance, the error should be distributed equally among these components(given some approximations) . The optimal number of T-gates obtained by using the global phase invariant distance is less. Furthermore, for approx-QFT the error due to pruning of rotation gates is less when measured in this distance.

scholarly journals Determining quantum phase diagrams of topological Kitaev-inspired models on NISQ quantum hardware

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 553
Author(s):  
Xiao Xiao ◽  
J. K. Freericks ◽  
A. F. Kemper
Keyword(s):  
Phase Diagrams ◽  
Fault Tolerant ◽  
Quantum Algorithms ◽  
Entanglement Entropy ◽  
Quantum Phase ◽  
Quantum Circuits ◽  
Quantum Computers ◽  
Quantum Model ◽  
Quantum Properties ◽  
Topological Quantum

Topological protection is employed in fault-tolerant error correction and in developing quantum algorithms with topological qubits. But, topological protection intrinsic to models being simulated, also robustly protects calculations, even on NISQ hardware. We leverage it by simulating Kitaev-inspired models on IBM quantum computers and accurately determining their phase diagrams. This requires constructing conventional quantum circuits for Majorana braiding to prepare the ground states of Kitaev-inspired models. The entanglement entropy is then measured to calculate the quantum phase boundaries. We show how maintaining particle-hole symmetry when sampling through the Brillouin zone is critical to obtaining high accuracy. This work illustrates how topological protection intrinsic to a quantum model can be employed to perform robust calculations on NISQ hardware, when one measures the appropriate protected quantum properties. It opens the door for further simulation of topological quantum models on quantum hardware available today.

scholarly journals Quantum Algorithms for Simulating the Lattice Schwinger Model

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 306 ◽  
Author(s):  
Alexander F. Shaw ◽  
Pavel Lougovski ◽  
Jesse R. Stryker ◽  
Nathan Wiebe
Keyword(s):  
Quantum Electrodynamics ◽  
Quantum Computer ◽  
Fault Tolerant ◽  
Quantum Algorithms ◽  
Gauge Field Theories ◽  
Coupling Constant ◽  
Schwinger Model ◽  
Pair Density ◽  
Trotter Formula ◽  
Better Than

The Schwinger model (quantum electrodynamics in 1+1 dimensions) is a testbed for the study of quantum gauge field theories. We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In particular, we perform a tight analysis of low-order Trotter formula simulations of the Schwinger model, using recently derived commutator bounds, and give upper bounds on the resources needed for simulations in both scenarios. In lattice units, we find a Schwinger model on N/2 physical sites with coupling constant x−1/2 and electric field cutoff x−1/2Λ can be simulated on a quantum computer for time 2xT using a number of T-gates or CNOTs in O~(N3/2T3/2xΛ) for fixed operator error. This scaling with the truncation Λ is better than that expected from algorithms such as qubitization or QDRIFT. Furthermore, we give scalable measurement schemes and algorithms to estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable–the mean pair density. Finally, we bound the root-mean-square error in estimating this observable via simulation as a function of the diamond distance between the ideal and actual CNOT channels. This work provides a rigorous analysis of simulating the Schwinger model, while also providing benchmarks against which subsequent simulation algorithms can be tested.

scholarly journals A multi-commodity network model for optimal quantum reversible circuit synthesis

PLoS ONE ◽  
2021 ◽  
Vol 16 (6) ◽  
pp. e0253140
Author(s):  
Jihye Jung ◽  
In-Chan Choi
Keyword(s):  
Quantum Computing ◽  
Network Flow ◽  
Fault Tolerant ◽  
Quantum Algorithms ◽  
Error Rates ◽  
Quantum Circuits ◽  
Circuit Synthesis ◽  
Multiple Control ◽  
Proposed Model ◽  
Implementation Costs

Quantum computing is a newly emerging computing environment that has recently attracted intense research interest in improving the output fidelity, fully utilizing its high computing power from both hardware and software perspectives. In particular, several attempts have been made to reduce the errors in quantum computing algorithms through the efficient synthesis of quantum circuits. In this study, we present an application of an optimization model for synthesizing quantum circuits with minimum implementation costs to lower the error rates by forming a simpler circuit. Our model has a unique structure that combines the arc-subset selection problem with a conventional multi-commodity network flow model. The model targets the circuit synthesis with multiple control Toffoli gates to implement Boolean reversible functions that are often used as a key component in many quantum algorithms. Compared to previous studies, the proposed model has a unifying yet straightforward structure for exploiting the operational characteristics of quantum gates. Our computational experiment shows the potential of the proposed model, obtaining quantum circuits with significantly lower quantum costs compared to prior studies. The proposed model is also applicable to various other fields where reversible logic is utilized, such as low-power computing, fault-tolerant designs, and DNA computing. In addition, our model can be applied to network-based problems, such as logistics distribution and time-stage network problems.

scholarly journals Transformed Structural Properties Method to Determine the Controllability and Observability of Robots

2021 ◽  
Vol 11 (7) ◽  
pp. 3082
Author(s):  
Dany Ivan Martinez ◽  
José de Jesús Rubio ◽  
Victor Garcia ◽  
Tomas Miguel Vargas ◽  
Marco Antonio Islas ◽  
...  
Keyword(s):  
Structural Properties ◽  
Approximation Error ◽  
Linearization Method ◽  
Controllability And Observability ◽  
Better Than

Many investigations use a linearization method, and others use a structural properties method to determine the controllability and observability of robots. In this study, we propose a transformed structural properties method to determine the controllability and observability of robots, which is the combination of the linearization and the structural properties methods. The proposed method uses a transformation in the robot model to obtain a linear robot model with the gravity terms and uses the linearization of the gravity terms to obtain the linear robot model; this linear robot model is used to determine controllability and observability. The described combination evades the structural conditions requirement and decreases the approximation error. The proposed method is better than previous methods because the proposed method can obtain more precise controllability and observability results. The modified structural properties method is compared with the linearization method to determine the controllability and observability of three robots.

How significant are the known collision and element distinctness quantum algorithms?

2004 ◽  
Vol 4 (3) ◽  
pp. 201-206
Author(s):  
L. Grover ◽  
T. Rudolph
Keyword(s):  
Search Algorithm ◽  
Quantum Algorithms ◽  
Search Space ◽  
Space Complexity ◽  
Quantum Search ◽  
Quantum Search Algorithm ◽  
Time Space ◽  
Simple Quantum ◽  
Structured Problems ◽  
Better Than

Quantum search is a technique for searching $N$ possibilities for a desired target in $O(\sqrt{N})$ steps. It has been applied in the design of quantum algorithms for several structured problems. Many of these algorithms require significant amount of quantum hardware. In this paper we propose the criterion that an algorithm which requires $O(S)$ hardware should be considered significant if it produces a speedup of better than $O\left(\sqrt{S}\right)$ over a simple quantum search algorithm. This is because a speedup of $O\left(\sqrt{S}\right)$ can be trivially obtained by dividing the search space into $S$ separate parts and handing the problem to $S$ independent processors that do a quantum search (in this paper we drop all logarithmic factors when discussing time/space complexity). Known algorithms for collision and element distinctness exactly saturate the criterion.

FAULT TOLERANT ROUTING IN HYPERCUBES AND STAR GRAPHS

1996 ◽  
Vol 06 (01) ◽  
pp. 127-136 ◽  
Author(s):  
QIAN-PING GU ◽  
SHIETUNG PENG
Keyword(s):  
Free Path ◽  
Fault Tolerant ◽  
Linear Time ◽  
Time Efficiency ◽  
Routing Problem ◽  
Star Graphs ◽  
Linear Time Algorithms ◽  
Better Than

In this paper, we give two linear time algorithms for node-to-node fault tolerant routing problem in n-dimensional hypercubes Hn and star graphs Gn. The first algorithm, given at most n−1 arbitrary fault nodes and two non-fault nodes s and t in Hn, finds a fault-free path s→t of length at most [Formula: see text] in O(n) time, where d(s, t) is the distance between s and t. Our second algorithm, given at most n−2 fault nodes and two non-fault nodes s and t in Gn, finds a fault-free path s→t of length at most d(Gn)+3 in O(n) time, where [Formula: see text] is the diameter of Gn. When the time efficiency of finding the routing path is more important than the length of the path, the algorithms in this paper are better than the previous ones.

scholarly journals Degenerate Quantum LDPC Codes With Good Finite Length Performance

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 585
Author(s):  
Pavel Panteleev ◽  
Gleb Kalachev
Keyword(s):  
Minimum Distance ◽  
Belief Propagation ◽  
Fault Tolerant ◽  
Large Family ◽  
Parity Check ◽  
Product Codes ◽  
Medium Length ◽  
Surface Code ◽  
Depolarizing Channel ◽  
Better Than

We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder.

scholarly journals Fast equivalence -- checking for quantum circuits

2010 ◽  
Vol 10 (9&10) ◽  
pp. 721-734
Author(s):  
Shigeru Yamashita ◽  
Igor L. Markov
Keyword(s):  
Formal Verification ◽  
Quantum Algorithms ◽  
Quantum Circuits ◽  
Equivalence Checking ◽  
Sat Solvers ◽  
Reversible Circuits ◽  
Verification Tools ◽  
Verification Methodology ◽  
Verification Techniques ◽  
Factoring Algorithm

We perform formal verification of quantum circuits by integrating several techniques specialized to particular classes of circuits. Our verification methodology is based on the new notion of a reversible miter that allows one to leverage existing techniques for simplification of quantum circuits. For reversible circuits which arise as runtime bottlenecks of key quantum algorithms, we develop several verification techniques and empirically compare them. We also combine existing quantum verification tools with the use of SAT-solvers. Experiments with circuits for Shor's number-factoring algorithm, containing thousands of gates, show improvements in efficiency by four orders of magnitude.

Optimizing Convolutional Neural Network Accelerator on Low-Cost FPGA

2021 ◽  
pp. 2150193
Author(s):  
Truong Quang Vinh ◽  
Dinh Viet Hai
Keyword(s):  
Neural Network ◽  
Convolutional Neural Network ◽  
Low Cost ◽  
Optimal Number ◽  
Data Reuse ◽  
Logic Element ◽  
Input Buffer ◽  
Classification Tasks ◽  
Processing Engine ◽  
Better Than

Convolutional neural network (CNN) is one of the most promising algorithms that outweighs other traditional methods in terms of accuracy in classification tasks. However, several CNNs, such as VGG, demand a huge computation in convolutional layers. Many accelerators implemented on powerful FPGAs have been introduced to address the problems. In this paper, we present a VGG-based accelerator which is optimized for a low-cost FPGA. In order to optimize the FPGA resource of logic element and memory, we propose a dedicated input buffer that maximizes the data reuse. In addition, we design a low resource processing engine with the optimal number of Multiply Accumulate (MAC) units. In the experiments, we use VGG16 model for inference to evaluate the performance of our accelerator and achieve a throughput of 38.8[Formula: see text]GOPS at a clock speed of 150[Formula: see text]MHz on Intel Cyclone V SX SoC. The experimental results show that our design is better than previous works in terms of resource efficiency.

scholarly journals Advanced exact synthesis of Clifford+T circuits

2020 ◽  
Vol 19 (9) ◽  
Author(s):  
Philipp Niemann ◽  
Robert Wille ◽  
Rolf Drechsler
Keyword(s):  
Large Scale ◽  
Fault Tolerant ◽  
Logic Synthesis ◽  
Research Problem ◽  
Quantum Systems ◽  
Quantum Circuits ◽  
Research Attention ◽  
Small Quantum ◽  
Significant Research ◽  
Important Research Problem

Abstract Quantum systems provide a new way of conducting computations based on the so-called qubits. Due to the potential for significant speed-ups, this field received significant research attention in recent years. The Clifford+T library is a very promising and popular gate library for these kinds of computations. Unlike other libraries considered so far, it consists of only a small number of gates for all of which robust, fault-tolerant realizations are known for many technologies that seem to be promising for large-scale quantum computing. As a consequence, (logic) synthesis of Clifford+T quantum circuits became an important research problem. However, previous work in this area has several drawbacks: Corresponding approaches are either only applicable to very small quantum systems or lead to circuits that are far from being optimal. The latter is mainly caused by the fact that current synthesis realizes the desired circuit by a local, i.e., column-wise, consideration of the underlying unitary transformation matrix to be synthesized. In this paper, we analyze the conceptual drawbacks of this approach and propose to overcome them by taking a global view of the matrices and perform a separation of concerns regarding individual synthesis steps. We precisely describe a corresponding algorithm as well as its efficient implementation on top of decision diagrams. Experimental results confirm the resulting benefits and show improvements of up to several orders of magnitudes in costs compared to previous work.

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