Permutations Of Qubits Research Articles

Quantum routing with teleportation

We study the problem of implementing arbitrary permutations of qubits under interaction constraints in quantum systems that allow for arbitrarily fast local operations and classical communication (LOCC). In particular, we show examples of speedups over swap-based and more general unitary routing methods by distributing entanglement and using LOCC to perform quantum teleportation. We further describe an example of an interaction graph for which teleportation gives a logarithmic speedup in the worst-case routing time over swap-based routing. We also study limits on the speedup afforded by quantum teleportation—showing an O(NlogN) upper bound on the separation in routing time for any interaction graph—and give tighter bounds for some common classes of graphs. Published by the American Physical Society 2024

Fold-Transversal Clifford Gates for Quantum Codes

We generalize the concept of folding from surface codes to CSS codes by considering certain dualities within them. In particular, this gives a general method to implement logical operations in suitable LDPC quantum codes using transversal gates and qubit permutations only.To demonstrate our approach, we specifically consider a [[30, 8, 3]] hyperbolic quantum code called Bring's code. Further, we show that by restricting the logical subspace of Bring's code to four qubits, we can obtain the full Clifford group on that subspace.

Hamiltonian simulation in the Pauli basis of multi-qubit clusters for condensed matter physics

We propose an efficient method for Hamiltonian simulation of multi-qubit quantum systems with special types of interaction. In our approach, the Hamiltonian of a \(n\)-qubit system should be represented as a linear combination of the standard Pauli basis operators, and then decomposed into a sum of partial Hamiltonians, which are, in general, not Pauli operators and satisfy some anticommutation relations. For three types of Hamiltonians, which are invariant with respect to permutations of qubits, the effectiveness of the main algorithm in the three-qubit cluster model is shown by calculating the operator exponentials for these Hamiltonians in an explicit analytical form. We also calculate the density operator, partition function, entropy, and free energy of the cluster weakly coupled to a thermal environment. In our model, the cluster is in the Gibbs state in the temperature interval \(0.1\!-\!2\:\!\text{K}\), which corresponds to the operating range of modern quantum processors. It follows from our analysis that the thermodynamic properties of such systems strongly depend on the type of internal interaction of qubits in the cluster.

Local unitary classes of states invariant under permutation subgroups

The study of entanglement properties of multi-qubit states that are invariant under permutations of qubits is motivated by potential applications in quantum computing, quantum communication, and quantum metrology. In this work, we generalize the notions of symmetrization, Dicke states, and the Majorana representation to the alternating, cyclic, and dihedral subgroups of the full group of permutations. We use these tools to characterize states that are invariant under these subgroups and analyze their entanglement properties.

Quantum routing with fast reversals.

We present methods for implementing arbitrary permutations of qubits under interaction constraints. Our protocols make use of previous methods for rapidly reversing the order of qubits along a path. Given nearest-neighbor interactions on a path of length , we show that there exists a constant such that the quantum routing time is at most , whereas any SWAP-based protocol needs at least time . This represents the first known quantum advantage over swAP-based routing methods and also gives improved quantum routing times for realistic architectures such as grids. Furthermore, we show that our algorithm approaches a quantum routing time of in expectation for uniformly random permutations, whereas SWAP-based protocols require time asymptotically. Additionally, we consider sparse permutations that route qubits and give algorithms with quantum routing time at most on paths and at most on general graphs with radius .

Hadamard-Free Circuits Expose the Structure of the Clifford Group

The Clifford group plays a central role in quantum randomized benchmarking, quantum tomography, and error correction protocols. Here we study the structural properties of this group. We show that any Clifford operator can be uniquely written in the canonical form F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> HSF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , where H is a layer of Hadamard gates, S is a permutation of qubits, and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> are parameterized Hadamard-free circuits chosen from suitable subgroups of the Clifford group. Our canonical form provides a one-to-one correspondence between Clifford operators and layered quantum circuits. We report a polynomial-time algorithm for computing the canonical form. We employ this canonical form to generate a random uniformly distributed n-qubit Clifford operator in runtime O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). The number of random bits consumed by the algorithm matches the information-theoretic lower bound. A surprising connection is highlighted between random uniform Clifford operators and the Mallows distribution on the symmetric group. The variants of the canonical form, one with a short Hadamard-free part and one allowing a circuit depth 9n implementation of arbitrary Clifford unitaries in the Linear Nearest Neighbor architecture are also discussed. Finally, we study computational quantum advantage where a classical reversible linear circuit can be implemented more efficiently using Clifford gates, and show an explicit example where such an advantage takes place.

Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits.

A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. Here we demonstrate that non-Abelian anyons in Turaev-Viro quantum error correcting codes can be moved over a distance of order of the code distance, and thus braided, by a constant depth local unitary quantum circuit followed by a permutation of qubits. Our gates are protected in the sense that the lengths of error strings do not grow by more than a constant factor. When applied to the Fibonacci code, our results demonstrate that a universal logical gate set can be implemented on encoded qubits through a constant depth unitary quantum circuit, and without increasing the asymptotic scaling of the space overhead. These results also apply directly to braiding of topological defects in surface codes. Our results reformulate the notion of braiding in general as an effectively instantaneous process, rather than as an adiabatic, slow process.

Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes

A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distancedthrough the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead ofO(d/log⁡d). For quantum circuits that allow parallel gate operations, this yields the optimal scaling of space-time overhead known to date.

CNOT Gate Optimizations via Qubit Permutations for IBM's Quantum Architectures

CNOT Gate Optimizations via Qubit Permutations for IBM's Quantum Architectures

Detection of genuine n-qubit entanglement via the proportionality of two vectors

In [Science 340, 1205, 7 June (2013)], via polytopes Michael Walter et al. proposed a sufficient condition detecting the genuinely entangled pure states. In this paper, we indicate that generally, the coefficient vector of a pure product state of $n$ qubits cannot be decomposed into a tensor product of two vectors, and show that a pure state of $n$ qubits is a product state if and only if there exists a permutation of qubits such that under the permutation, its coefficient vector arranged in ascending lexicographical order can be decomposed into a tensor product of two vectors. The contrapositive of this result reads that a pure state of $n$ qubits is genuinely entangled if and only if its coefficient vector cannot be decomposed into a tensor product of two vectors under any permutation of qubits. Further, by dividing a coefficient vector into $2^{i}$ equal-size block vectors, we show that the coefficient vector can be decomposed into a tensor product of two vectors if and only if any two non-zero block vectors of the coefficient vector are proportional. In terms of \textquotedblleft proportionality\textquotedblright , we can rephrase that a pure state of $n$ qubits is genuinely entangled if and only if there are two non-zero block vectors of the coefficient vector which are not proportional under any permutation of qubits. Thus, we avoid decomposing a coefficient vector into a tensor product of two vectors to detect the genuine entanglement. We also present the full decomposition theorem for product states of n qubits.

Noisy frequency estimation with noisy probes

We consider frequency estimation in a noisy environment with noisy probes. This builds on previous studies, most of which assume that the initial probe state is pure, while the encoding process is noisy, or that the initial probe state is mixed, while the encoding process is noiseless. Our work is more representative of reality, where noise is unavoidable in both the initial state of the probe and the estimation process itself. We prepare the probe in a GHZ diagonal state, starting from n + 1 qubits in an arbitrary uncorrelated mixed state, and subject it to parameter encoding under dephasing noise. For this scheme, we derive a simple formula for the (quantum and classical) Fisher information, and show that quantum enhancements do not depend on the initial mixedness of the qubits. That is, we show that the so-called ‘Zeno’ scaling is attainable when the noise present in the encoding process is time inhomogeneous. This scaling does not depend on the mixedness of the initial probe state, and it is retained even for highly mixed states that can never be entangled. We then show that the sensitivity of the probe in our protocol is invariant under permutations of qubits, and monotonic in purity of the initial state of the probe. Finally, we discuss two limiting cases, where purity is either distributed evenly among the probes or concentrated in a single probe.

Orthogonal product bases of four qubits

An orthogonal product basis (OPB) of a finite-dimensional Hilbert space is an orthonormal basis of consisting of product vectors . We show that the problem of constructing the OPBs of an n-qubit system can be reduced to a purely combinatorial problem. We solve this combinatorial problem in the case of four qubits and obtain 33 multiparameter families of OPBs. Each OPB of four qubits is equivalent, under local unitary operations and qubit permutations, to an OPB belonging to at least one of these families.

Number of superclasses of four-qubit entangled states under the inductive entanglement classification

Lamata et al. use an inductive approach to classify the entangled pure states of four qubits under stochastic local operations and classical communication (SLOCC) [PRA 75(2), 022318 (2007)]. The inductive method yields a priori ten different entanglement superclasses, of which they discard three as empty. One of the remaining superclasses is split in two, resulting in eight superclasses of genuine four-qubit entanglement. Here, we show that two of the three discarded superclasses are in fact non-empty and should have been retained. We give explicit expressions for the canonical states for those superclasses, up to SLOCC and qubit permutations. Furthermore, we confirm that the third discarded superclass is indeed empty, yielding a total of ten superclasses of genuine four-qubit entanglement under the inductive classification scheme.

Canonical form of three-fermion pure-states with six single particle states

We construct a canonical form for pure states in \documentclass[12pt]{minimal}\begin{document}$\mathop {\wedge }\nolimits ^3({\mathbb {C}}^6)$\end{document}∧3(C6), the three-fermion system with six single particle states, under local unitary (LU) transformations, i.e., the unitary group U(6). We also construct a minimal set of generators of the algebra of polynomial U(6)-invariants on \documentclass[12pt]{minimal}\begin{document}$\mathop {\wedge }\nolimits ^3({\mathbb {C}}^6)$\end{document}∧3(C6). It turns out that this algebra is isomorphic to the algebra of polynomial LU-invariants of three-qubits which are additionally invariant under qubit permutations. As a consequence of this surprising fact, we deduce that there is a one-to-one correspondence between the U(6)-orbits of pure three-fermion states in \documentclass[12pt]{minimal}\begin{document}$\mathop {\wedge }\nolimits ^3({\mathbb {C}}^6)$\end{document}∧3(C6) and the LU orbits of pure three-qubit states when qubit permutations are allowed. As an important byproduct, we obtain a new canonical form for pure three-qubit states under LU transformations U(2) × U(2) × U(2) (no qubit permutations allowed).

SLOCC determinant invariants of order 2n/2 for even n qubits

In this paper, we study SLOCC determinant invariants of order 2n/2 for any even n qubits, which satisfy the SLOCC determinant equations. The determinant invariants can be constructed by a simple method and the set of all these determinant invariants is complete with respect to permutations of qubits. The SLOCC entanglement classification can be achieved via the vanishing or not of the determinant invariants. We exemplify the method for several even number of qubits, with an emphasis on six qubits.

On polynomial invariants of several qubits

It is a recent observation that entanglement classification for qubits is closely related to local SL(2,C)-invariants including the invariance under qubit permutations [Dür, et al., Phys. Rev. A 62, 062314 (2000); Osterloh, A. and Siewert, J., Phys. Rev. A 72, 012337 (2005); O. Chterental and D. Ž. Ðoković, Linear Algebra Research Advances (Nova Science, Hauppauge, N.Y., 2007), Chap. 4, p. 133], which has been termed SL∗ invariance. In order to single out the SL∗ invariants, we analyze the SL(2,C)-invariants of four (five) qubits and decompose them into irreducible modules for the symmetric group S4 (S5) of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of SL∗-invariants vanishing on arbitrary product states. We find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced by Osterloh and Siewert [Phys. Rev. A 72, 012337 (2005); Int. J. Quant. Inf. 4, 531 (2006)]. Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the Ω-process [Luque, J.-G. and Thibon, J.-Y., J. Phys. A 39, 371 (2005)]. As the degrees of invariants increase, the alternative method proves to be particularly efficient.

Scalable protocol for identification of correctable codes

The task of finding a correctable encoding that protects against some physical quantum process is in general hard. Two main obstacles are that an exponential number of experiments are needed to gain complete information about the quantum process, and known algorithmic methods for finding correctable encodings involve operations on exponentially large matrices. However, we show that in some cases it is possible to find such encodings with only partial information about the quantum process. Such useful partial information can be systematically extracted by averaging the channel under the action of a set of unitaries in a process known as "twirling". In this paper we prove that correctable encodings for a twirled channel are also correctable for the original channel. We investigate the particular case of twirling over the set of Pauli operators and qubit permutations, and show that the resulting quantum operation can be characterized experimentally in a scalable manner. We also provide a postprocessing scheme for finding unitarily correctable codes for these twirled channels which does not involve exponentially large matrices.

Symmetrizing evolutions

We introduce quantum procedures for making G -invariant the dynamics of an arbitrary quantum system S, where G is a finite group acting on the space state of S. Several applications of this idea are discussed. In particular when S is a N-qubit quantum computer interacting with its environment and G the symmetric group of qubit permutations, the resulting effective dynamics admits noiseless subspaces. Moreover it is shown that the recently introduced iterated-pulses schemes for reducing decoherence in quantum computers fit in this general framework. The noise-inducing component of the Hamiltonian is filtered out by the symmetrization procedure just due to its transformation properties.