scholarly journals Schur–Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations

2021 ◽  
Author(s):  
David Gross ◽  
Sepehr Nezami ◽  
Michael Walter
Keyword(s):  
Quantum Information ◽  
Duality Theory ◽  
Fault Tolerant ◽  
Property Testing ◽  
List Type ◽  
Clifford Group ◽  
Weyl Duality ◽  
De Finetti ◽  
Tensor Powers ◽  
Stabilizer States

AbstractSchur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t-fold tensor powers $$U^{\otimes t}$$ U ⊗ t of all unitaries $$U\in U(d)$$ U ∈ U ( d ) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications: We resolve an open problem in quantum property testing by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group. We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well). We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) – a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem. We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.

scholarly journals Transforming graph states using single-qubit operations

2018 ◽  
Vol 376 (2123) ◽  
pp. 20170325 ◽  
Author(s):  
Axel Dahlberg ◽  
Stephanie Wehner
Keyword(s):  
Quantum Information ◽  
Error Correcting Codes ◽  
Graph States ◽  
Target State ◽  
Single Qubit ◽  
Minor Problem ◽  
Fixed Parameter ◽  
Source State ◽  
Quantum Error ◽  
Stabilizer States

Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. What is more, we provide a recipe to obtain the sequence of LC+LPM+CC operations which prepare the desired target state from the source state, and show how these operations can be applied in parallel to reach the target state in constant time. Our algorithm has applications in quantum networks, quantum computing, and can also serve as a design tool—for example, to find transformations between quantum error correcting codes. We provide a software implementation of our algorithm that makes this tool easier to apply. A key insight leading to our algorithm is to show that the problem is equivalent to one in graph theory, which is to decide whether some graph G ′ is a vertex-minor of another graph G . The vertex-minor problem is, in general, -Complete, but can be solved efficiently on graphs which are not too complex. A measure of the complexity of a graph is the rank-width which equals the Schmidt-rank width of a subclass of stabilizer states called graph states, and thus intuitively is a measure of entanglement. Here, we show that the vertex-minor problem can be solved in time O (| G | 3 ), where | G | is the size of the graph G , whenever the rank-width of G and the size of G ′ are bounded. Our algorithm is based on techniques by Courcelle for solving fixed parameter tractable problems, where here the relevant fixed parameter is the rank width. The second half of this paper serves as an accessible but far from exhausting introduction to these concepts, that could be useful for many other problems in quantum information. This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.

scholarly journals Engineering the quantum-classical interface of solid-state qubits

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
David J Reilly
Keyword(s):  
Quantum Information ◽  
Solid State ◽  
Degrees Of Freedom ◽  
Technology Development ◽  
Fault Tolerant ◽  
Building Blocks ◽  
Experimental Physicist ◽  
And Mathematics ◽  
Classical Control ◽  
And Control

AbstractSpanning a range of hardware platforms, the building-blocks of quantum processors are today sufficiently advanced to begin work on scaling-up these systems into complex quantum machines. A key subsystem of all quantum machinery is the interface between the isolated qubits that encode quantum information and the classical control and readout technology needed to operate them. As few-qubit devices are combined to construct larger, fault-tolerant quantum systems in the near future, the quantum-classical interface will pose new challenges that increasingly require approaches from the engineering disciplines in combination with continued fundamental advances in physics, materials and mathematics. This review describes the subsystems comprising the quantum-classical interface from the viewpoint of an engineer, experimental physicist or student wanting to enter the field of solid-state quantum information technology. The fundamental signalling operations of readout and control are reviewed for a variety of qubit platforms, including spin systems, superconducting implementations and future devices based on topological degrees-of-freedom. New engineering opportunities for technology development at the boundary between qubits and their control hardware are identified, transversing electronics to cryogenics.

scholarly journals Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography

2017 ◽  
Vol 8 (1) ◽  
Author(s):  
Robin Blume-Kohout ◽  
John King Gamble ◽  
Erik Nielsen ◽  
Kenneth Rudinger ◽  
Jonathan Mizrahi ◽  
...  
Keyword(s):  
Quantum Information ◽  
Fault Tolerance ◽  
Error Correction ◽  
Error Rate ◽  
Fault Tolerant ◽  
Fast Algorithms ◽  
Quantum Error Correction ◽  
Qubit Operations ◽  
Quantum Error ◽  
Qubit Operation

Abstract Quantum information processors promise fast algorithms for problems inaccessible to classical computers. But since qubits are noisy and error-prone, they will depend on fault-tolerant quantum error correction (FTQEC) to compute reliably. Quantum error correction can protect against general noise if—and only if—the error in each physical qubit operation is smaller than a certain threshold. The threshold for general errors is quantified by their diamond norm. Until now, qubits have been assessed primarily by randomized benchmarking, which reports a different error rate that is not sensitive to all errors, and cannot be compared directly to diamond norm thresholds. Here we use gate set tomography to completely characterize operations on a trapped-Yb+-ion qubit and demonstrate with greater than 95% confidence that they satisfy a rigorous threshold for FTQEC (diamond norm ≤6.7 × 10−4).

Computational model underlying the one-way quantum computer

2002 ◽  
Vol 2 (6) ◽  
pp. 443-486
Author(s):  
R. Raussendorf ◽  
H. Briegel
Keyword(s):  
Information Processing ◽  
Quantum Information ◽  
Computational Model ◽  
Quantum Logic ◽  
Quantum Information Processing ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Single Step ◽  
Clifford Group ◽  
The One

In this paper we present the computational model underlying the one-way quantum computer which we introduced recently [Phys. Rev. Lett. {\bf{86}}, 5188 (2001)]. The one-way quantum computer has the property that any quantum logic network can be simulated on it. Conversely, not all ways of quantum information processing that are possible with the one-way quantum computer can be understood properly in network model terms. We show that the logical depth is, for certain algorithms, lower than has so far been known for networks. For example, every quantum circuit in the Clifford group can be performed on the one-way quantum computer in a single step.

scholarly journals Schur–Weyl duality and the product of randomly-rotated symmetries by a unitary Brownian motion

2021 ◽  
pp. 2150002
Author(s):  
Nizar Demni ◽  
Tarek Hamdi
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Hermitian Matrix ◽  
Direct Proof ◽  
Fixed Time ◽  
Stochastic Calculus ◽  
Unitary Matrix ◽  
Weyl Duality ◽  
Jacobi Process ◽  
Tensor Powers

In this paper, we introduce and study a unitary matrix-valued process which is closely related to the Hermitian matrix-Jacobi process. It is precisely defined as the product of a deterministic self-adjoint symmetry and a randomly-rotated one by a unitary Brownian motion. Using stochastic calculus and the action of the symmetric group on tensor powers, we derive an ordinary differential equation for the moments of its fixed-time marginals. Next, we derive an expression of these moments which involves a unitary bridge between our unitary process and another independent unitary Brownian motion. This bridge motivates and allows to write a second direct proof of the obtained moment expression.

Graphical algorithms and threshold error rates for the 2d color code

2010 ◽  
Vol 10 (9&10) ◽  
pp. 780-802
Author(s):  
David S. Wang ◽  
Austin G. Fowler ◽  
Charles D. Hill ◽  
Lloyd C.L. Hollenberg
Keyword(s):  
Quantum Computation ◽  
Error Rate ◽  
Graph Matching ◽  
Fault Tolerant ◽  
Error Rates ◽  
Color Code ◽  
Clifford Group ◽  
Universal Quantum ◽  
Surface Code ◽  
Topological Error

Recent work on fault-tolerant quantum computation making use of topological error correction shows great potential, with the 2d surface code possessing a threshold error rate approaching 1\%. However, the 2d surface code requires the use of a complex state distillation procedure to achieve universal quantum computation. The color code of is a related scheme partially solving the problem, providing a means to perform all Clifford group gates transversally. We review the color code and its error correcting methodology, discussing one approximate technique based on graph matching. We derive an analytic lower bound to the threshold error rate of 6.25\% under error-free syndrome extraction, while numerical simulations indicate it may be as high as 13.3\%. Inclusion of faulty syndrome extraction circuits drops the threshold to approximately 0.10 \pm 0.01\%.

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