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Towards a Minimal Stabilizer ZX-calculus [article]

Miriam Backens, Simon Perdrix, Quanlong Wang
2020 arXiv   pre-print
The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics.  ...  We previously showed that the stabilizer ZX-calculus can be simplified by reducing the number of rewrite rules, without losing the property of completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017  ...  This approach is not inconsistent with the notion of working towards a minimal version of the stabilizer zx-calculus: we are looking for a version of the calculus where all rewrite rules are provably necessary  ... 
arXiv:1709.08903v6 fatcat:q2gm674snval7bkohtlxh2qayi

Towards a Minimal Stabilizer ZX-calculus

Miriam Backens, Simon Perdrix, Quanlong Wang
2017 Logical Methods in Computer Science  
The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics.  ...  We previously showed that the stabilizer ZX-calculus can be simplified by reducing the number of rewrite rules, without losing the property of completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017  ...  This approach is not inconsistent with the notion of working towards a minimal version of the stabilizer zx-calculus: we are looking for a version of the calculus where all rewrite rules are provably necessary  ... 
doi:10.23638/lmcs-16(4:19)2020 fatcat:qg62mqnybrhpjl3yaldlsqgsbe

Circuit Relations for Real Stabilizers: Towards TOF+H [article]

Cole Comfort
2019 arXiv   pre-print
The real stabilizer fragment of quantum mechanics was shown to have a complete axiomatization in terms of the angle-free fragment of the ZX-calculus.  ...  We then construct translations to and from the angle-free fragment of the ZX-calculus, showing that they are inverses.  ...  However, we are interested in a simple fragment of the ZX-calculus, namely the angle-free calculus for real stabilizer circuits, ZX π , described in [15] (slightly modified to account for scalars): Definition  ... 
arXiv:1904.10614v2 fatcat:5osaxj3wnvcptbw6w34rdmflqq

Completeness of the Phase-free ZH-calculus [article]

John van de Wetering, Sal Wolffs
2019 arXiv   pre-print
We present a modified rule-set for the phase-free ZH-calculus and show that it is complete. We further discuss the minimality of this rule-set and we give an intuitive interpretation of the rules.  ...  The ZH-calculus is a graphical calculus for linear maps between qubits that allows a natural representation of the Toffoli+Hadamard gate set.  ...  Whereas the ZX-calculus allows easy representation of stabilizer and phase gates, the ZH-calculus allows one to easily represent the Hadamard+Toffoli gate set.  ... 
arXiv:1904.07545v1 fatcat:rbax3qrw5vff7fc5shnyu2xqre

Graphical Calculi and their Conjecture Synthesis [article]

Hector Miller-Bakewell
2020 arXiv   pre-print
The second is the calculus ZQ, an edge-decorated calculus which naturally expresses arbitrary qubit rotations, eliminating the need for non-linear rules such as (EU) of ZX.  ...  To this process we introduce a generalisation step, which itself requires the ability to infer and then verify parameterised families of theorems.  ...  [Bac15] 2015 Making the stabilizer ZX-calculus complete for scalars [BPW17] 2017 A Simplified Stabilizer ZX-calculus [Jea+17] 2017 ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford  ... 
arXiv:2010.03914v1 fatcat:z4k2so32hvfjfjismyzwjjqmey

Numerical Analysis of a Hydrodynamic Herringbone Grooved Journal Bearing

D. Souchet, A. Senouci, H. Zaidi, M. Amirat, Chuan-Yu Wu
2020 Journal of Engineering  
The researches made before on these types of groove bearing have shown that they present a good dynamical behavior with a low eccentricity and a low axial flow.  ...  In this paper, a numerical study of a herringbone journal bearing operating behavior, under laminar and isothermal regime, is presented.  ...  Calculus of the Herringbone Grooved Bearings Stability.  ... 
doi:10.1155/2020/3796214 fatcat:loztawqu7jbnxby6cwyzs7dqiu

Efficient magic state factories with a catalyzed |CCZ⟩ to 2|T⟩ transformation

Craig Gidney, Austin G. Fowler
2019 Quantum  
It has a footprint 25% smaller than the factory in \cite{fowler2018} but outputs |T⟩ states twice as quickly.  ...  Assuming a physical gate error rate of 10−3, our CCZ factory can produce ∼1010 states on average before an error occurs.  ...  Campbell for reviewing a draft of this paper and suggesting useful references.  ... 
doi:10.22331/q-2019-04-30-135 fatcat:xm7eibf2fzew3opquxcbzgmfey

Page 1785 of Mathematical Reviews Vol. , Issue 86d [page]

1986 Mathematical Reviews  
Author summary: “In this paper, we study the global stability of a generalized predator-prey system dz/dt = rg(x)—yp(zx), dy/dt = y|cp(x) —q(z)|, and give an example.”  ...  There has been a need for high level probabilistic work directed toward neural learning in pattern classification.  ... 

Parameter Optimization of Droop Controllers for Microgrids in Islanded Mode by the SQP Method with Gradient Sampling

Peijie Li, Ziyi Yang, Shuchen Huang, Jun Zhang, Chun Wei
2021 Complexity  
For enhancing the stability of the microgrid operation, this paper proposes an optimization model considering the small-signal stability constraint.  ...  Due to the nonsmooth property of the spectral abscissa function, the droop controller parameters' optimization is a nonsmooth optimization problem.  ...  It is a mathematical eigenvalue derived at an equilibrium point; and the eigenvalue sensitivity with respect to variable x can be expressed as zλ i zx � ψ ⊤ i z A mg /zx ϕ i ψ ⊤ i ϕ i , (56) where A mg  ... 
doi:10.1155/2021/5634274 fatcat:klck5zv3ufewlflodrzawnie6m

New Algorithms for Polynomial J-Spectral Factorization

Harry L. Trentelman, Paolo Rapisarda
1999 MCSS. Mathematics of Control, Signals and Systems  
These algorithms are based on the calculus of twovariable polynomial matrices and associated quadratic di¨erential forms, and share the common feature that the problem is lifted from the original onevariable  ...  For this, we propose an algorithm that uses the notion of a Pick matrix associated with a given two-variable polynomial matrix.  ...  The correspondence between QDFs and two-variable polynomial matrices allows us to develop a calculus that has applications in stability theory, optimal control, and H y -control (see [19] and [16] )  ... 
doi:10.1007/pl00009844 fatcat:4xj3qckawjdq3jcjiqkfcfant4

There and back again: A circuit extraction tale

Miriam Backens, Hector Miller-Bakewell, Giovanni de Felice, Leo Lobski, John van de Wetering
2021 Quantum  
We bring together several known rewrite rules for measurement patterns and formalise them in a unified notation using the ZX-calculus.  ...  One-way computations are represented using the ZX-calculus, hence the algorithm also represents the most general known procedure for extracting circuits from ZX-diagrams.  ...  The ZX-calculus is universal in the sense that any linear map can be expressed as a ZX-diagram.  ... 
doi:10.22331/q-2021-03-25-421 fatcat:puoh2jnwibgwxlxuksteylvz3q

There and back again: A circuit extraction tale [article]

Miriam Backens, Hector Miller-Bakewell, Giovanni de Felice, Leo Lobski, John van de Wetering
2021 arXiv   pre-print
We bring together several known rewrite rules for measurement patterns and formalise them in a unified notation using the ZX-calculus.  ...  One-way computations are represented using the ZX-calculus, hence the algorithm also represents the most general known procedure for extracting circuits from ZX-diagrams.  ...  The ZX-calculus is universal in the sense that any linear map can be expressed as a ZX-diagram.  ... 
arXiv:2003.01664v3 fatcat:lkfzysa2sjb37f7vtaiuvdc4om

Generalized Snell's Law for Weighted Minimal Surface in Heterogeneous Media

Zhilin Li, Xiaobiao Lin, Monica Torres, Hongkai Zhao
2003 Methods and Applications of Analysis  
The solution to the weighted minimal surface problem is continuous but the derivatives have a jump across the interface where the medium property is discontinuous.  ...  The weighted minimal surface problem in heterogeneous media is studied in this paper.  ...  Mathematically, the case when c = 0 has been studied in [17] using standard arguments of the calculus of variations.  ... 
doi:10.4310/maa.2003.v10.n2.a3 fatcat:7wvanxfuyvcbhpg2ph3xw6ybbq

A Prediction-Correction Algorithm for Real-Time Model Predictive Control [article]

Santiago Paternain, Manfred Morari, Alejandro Ribeiro
2019 arXiv   pre-print
a Hessian.  ...  In this work we adapt a prediction-correction algorithm for continuous time-varying convex optimization problems to solve dynamic programs arising from Model Predictive Control.  ...  INPUT TO STATE STABILITY We work next towards establishing input to state stability of the system (16) with respect to the approximation errors.  ... 
arXiv:1911.10051v1 fatcat:674f2d5lvzcm5bpiyd2topq6ne

ZX-Calculus and Extended Wolfram Model Systems II: Fast Diagrammatic Reasoning with an Application to Quantum Circuit Simplification [article]

Jonathan Gorard, Manojna Namuduri, Xerxes D. Arsiwalla
2021 arXiv   pre-print
system, before proceeding to demonstrate how to embed the higher-order logic of the ZX-calculus rules within this first-order equational framework.  ...  especially upon the application of this new algorithm to the problem of automated circuit simplification in quantum information theory, using Wolfram model multiway operator systems combined with the ZX-calculus  ...  We also present a fully worked example of an automatically-generated proof of correctness for a simple quantum teleportation protocol in the ZX-calculus, illustrating explicitly how the individual ZX-calculus  ... 
arXiv:2103.15820v1 fatcat:u6ov7udmqna5bngudx6tbt7fnu
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