Classification of Schmidt-rank-two multipartite unitary gates by singular number

• URL: http://arxiv.org/abs/2208.09604v1
• Date: Sat, 20 Aug 2022 04:22:12 GMT
• Title: Classification of Schmidt-rank-two multipartite unitary gates by singular number
• Authors: Yi Shen and Lin Chen and Li Yu
• Abstract summary: We investigate the classification of genuine multipartite unitary gates of Schmidt rank two. For genuine multipartite unitary gates of Schmidt rank two, there is an essential fact that their Schmidt decompositions are unique. We show that a three-qubit diagonal unitary gate has Schmidt rank at most three, and present a necessary and sufficient condition for such a unitary gate of Schmidt rank three.
• Score: 12.414561605185497
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The multipartite unitary gates are called genuine if they are not product unitary operators across any bipartition. We mainly investigate the classification of genuine multipartite unitary gates of Schmidt rank two, by focusing on the multiqubit scenario. For genuine multipartite (excluding bipartite) unitary gates of Schmidt rank two, there is an essential fact that their Schmidt decompositions are unique. Based on this fact, we propose a key notion named as singular number to classify the unitary gates concerned. The singular number is defined as the number of local singular operators in the Schmidt decomposition. We then determine the accurate range of singular number. For each singular number, we formulate the parametric Schmidt decompositions of genuine multiqubit unitary gates under local equivalence. Finally, we extend the study to three-qubit diagonal unitary gates due to the close relation between diagonal unitary gates and Schmidt-rank-two unitaries. We start with discussing two typical examples of Schmidt rank two, one of which is a fundamental three-qubit unitary gate, i.e., the CCZ gate. Then we characterize the diagonal unitary gates of Schmidt rank greater than two. We show that a three-qubit diagonal unitary gate has Schmidt rank at most three, and present a necessary and sufficient condition for such a unitary gate of Schmidt rank three. This completes the characterization of all genuine three-qubit diagonal unitary gates.

Related papers

• Multipartite entangling power by von Neumann entropy [6.193378336510727]
  Quantifying entanglement generation of a multipartite unitary operation is a key problem in quantum information processing. We analytically derive the entangling power of Schmidt-rank-two multi-qubit unitary operations by the minimal convex sum of modulo-one complex numbers. We investigate the widely-used multi-qubit gates, for example, the entangling and assisted entangling power of the $n$-qubit Toffoli gate is one ebit.
  arXiv  Detail & Related papers  (2024-10-20T02:18:36Z)
• A framework for semi-universality: Semi-universality of 3-qudit SU(d)-invariant gates [0.8192907805418583]
  We show that 3-qudit SU(d)-invariant gates are semi-universal, i.e., generate all SU(d)-invariant unitaries. We also prove that these gates achieve full universality when supplemented with 3 ancilla qudits. In particular, we establish that even though 2-qudit SU(d)-invariant gates are not themselves semi-universal, they become universal with at most 11 ancilla qudits.
  arXiv  Detail & Related papers  (2024-07-30T23:54:46Z)
• Quantum control landscape for generation of $H$ and $T$ gates in an open qubit with both coherent and environmental drive [57.70351255180495]
  An important problem in quantum computation is generation of single-qubit quantum gates such as Hadamard ($H$) and $pi/8$ ($T$) Here we consider the problem of optimal generation of $H$ and $T$ gates using coherent control and the environment as a resource acting on the qubit via incoherent control.
  arXiv  Detail & Related papers  (2023-09-05T09:05:27Z)
• The Schmidt rank for the commuting operator framework [58.720142291102135]
  The Schmidt rank is a measure for the entanglement dimension of a pure bipartite state. We generalize the Schmidt rank to the commuting operator framework. We analyze bipartite states and compute the Schmidt rank in several examples.
  arXiv  Detail & Related papers  (2023-07-21T14:37:33Z)
• A vertical gate-defined double quantum dot in a strained germanium double quantum well [48.7576911714538]
  Gate-defined quantum dots in silicon-germanium heterostructures have become a compelling platform for quantum computation and simulation. We demonstrate the operation of a gate-defined vertical double quantum dot in a strained germanium double quantum well. We discuss challenges and opportunities and outline potential applications in quantum computing and quantum simulation.
  arXiv  Detail & Related papers  (2023-05-23T13:42:36Z)
• Software mitigation of coherent two-qubit gate errors [55.878249096379804]
  Two-qubit gates are important components of quantum computing. But unwanted interactions between qubits (so-called parasitic gates) can degrade the performance of quantum applications. We present two software methods to mitigate parasitic two-qubit gate errors.
  arXiv  Detail & Related papers  (2021-11-08T17:37:27Z)
• Universality for Sets of Three-Valued Qubit Gates [0.0]
  We will do a review of the theory behind some essential proofs of quantum gate universality for qubit gates. We will then show a new way of extending those statements to arbitrary qutrit gates in an analogous manner.
  arXiv  Detail & Related papers  (2021-09-10T22:22:54Z)
• Quantum simulation of $\phi^4$ theories in qudit systems [53.122045119395594]
  We discuss the implementation of quantum algorithms for lattice $Phi4$ theory on circuit quantum electrodynamics (cQED) system. The main advantage of qudit systems is that its multi-level characteristic allows the field interaction to be implemented only with diagonal single-qudit gates.
  arXiv  Detail & Related papers  (2021-08-30T16:30:33Z)
• Random quantum circuits anti-concentrate in log depth [118.18170052022323]
  We study the number of gates needed for the distribution over measurement outcomes for typical circuit instances to be anti-concentrated. Our definition of anti-concentration is that the expected collision probability is only a constant factor larger than if the distribution were uniform. In both the case where the gates are nearest-neighbor on a 1D ring and the case where gates are long-range, we show $O(n log(n)) gates are also sufficient.
  arXiv  Detail & Related papers  (2020-11-24T18:44:57Z)
• Constructing three-qubit unitary gates in terms of Schmidt rank and CNOT gates [6.127256542161883]
  We explicitly construct the gates of Schmidt rank from one to seven. We implement the gates using quantum circuits of CNOT gates and local Hadamard and flip gates.
  arXiv  Detail & Related papers  (2020-07-06T01:16:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.