We study the polynomial functions on tensor states in $(C^n)^{\otimes k}$ which are invariant under $SU(n)^k$. We describe the space of invariant polynomials in terms of symmetric group representations. For $k$ even, the smallest degree for invariant polynomials is $n$ and in degree $n$ we find a natural generalization of the determinant. For $n,d$ fixed, we describe the asymptotic behavior of the dimension of the space of invariants as $k\to\infty$. We study in detail the space of homogeneous degree 4 invariant polynomial functions on $(C^2)^{\otimes k}$.
Protection of quantum information from noise is a massive challenge. One avenue people have begun to explore is reducing the number of particles needing to be protected from noise and instead use systems with more states, so called qudit quantum computers. These systems will require codes which utilize the full computational space. Many prior qudit codes are very restrictive on relations between parameters. In this paper we show that codes for these systems can be derived from already known codes, often relaxing the constraints somewhat, a result which could prove to be very useful for fault-t…
A permutation-invariant quantum code on $N$ qudits is any subspace stabilized by the matrix representation of the symmetric group $S_N$ as permutation matrices that permute the underlying $N$ subsystems. When each subsystem is a complex Euclidean space of dimension $q \ge 2$, any permutation-invariant code is a subspace of the symmetric subspace of $(\mathbb C^q)^N.$ We give an algebraic construction of new families of of $d$-dimensional permutation-invariant codes on at least $(2t+1)^2(d-1)$ qudits that can also correct $t$ errors for $d \ge 2$. The construction of our codes relies on a real …
Information is encoded in a qubit in the form of its Bloch vector. In this paper, we propose protocols for remote transfers of information in a known and an unknown qubit to qudits using $SU(2)$- invariant $\frac{1}{2}\otimes S$ discordant states as a channel. These states have been identified as separable equivalents of the two-qubit entangled Werner states in [Bharath & Ravishankar, Phys. Rev. A 89, 062110]. Due to $SU(2) \times SU(2)$ invariance of these states, the remote qudit can be changed by performing appropriate measurements on the qubit. We also propose a protocol for transferring i…
We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with code distance being the linear system size, is decomposed by a local Clifford circuit of constant depth into a finite number of copies of the toric code (abelian discrete gauge theory) stabilizer group. This means that under local Clifford circuits the number of toric code copies is the complete invariant of topological Pauli stabilizer codes. Previously, the same conclusion was obtained under the assumption of nonchirality for qubit cod…
In this work, we propose three-dimensional photonic circuit designs that guarantee a considerable reduction in the complexity of circuits for the purpose of performing quantum state tomography of N-dimensional path qudits. The POVM (Positive Operator-Valued Measure) chosen in this work ensures that, for odd dimensions, such process is minimal. Our proposal consists of organizing the waveguides that form the circuit as a square array formed by N vertical sectors composed of N waveguides each, arranged in the vertical direction. Based on the symmetry of the chosen POVM, the interferometer acting…
Permutation-invariant codes: a numerical study and qudit constructions
QuSoft, Amsterdam（QuSoft）；Institute of Physics, University of Amsterdam（阿姆斯特丹大学物理研究所）；Institute for Logic, Language and Computation, University of Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）；Centre for the Mathematics of Quantum Theory, University of Copenhagen（哥本哈根大学量子理论数学中心）
One hurdle to performing reliable quantum computations is overcoming noise. One possibility is to reduce the number of particles needing to be protected from noise and instead use systems with more states, so called qudit quantum computers. In this paper we show that codes for these systems can be derived from already known codes, and in particular that degenerate stabilizer codes can have their distance also promised upon sufficiently large local-dimension, as well as a new bound on the local-dimension required to preserve the distance of local-dimension-invariant codes, which is a result whi…
Building an efficient quantum memory in high-dimensional Hilbert spaces is one of the fundamental requirements for establishing high-dimensional quantum repeaters, where it offers many advantages over two-dimensional quantum systems, such as a larger information capacity and enhanced noise resilience. To date, there have been no reports about how to achieve an efficient high-dimensional quantum memory. Here, we experimentally realize a quantum memory that is operational in Hilbert spaces of up to 25 dimensions with a storage efficiency of close to 60%. The proposed approach exploits the spatia…
Local symmetric quantum circuits provide a simple framework to study the dynamics and phases of complex quantum systems with conserved charges. However, some of their basic properties have not yet been understood. Recently, it has been shown that such quantum circuits only generate a restricted subset of symmetric unitary transformations [I. Marvian, Nature Physics, 2022]. In this paper, we consider circuits with 2-local SU(d)-invariant unitaries acting on qudits, i.e., d-dimensional quantum systems. Our results reveal a significant distinction between the cases of d = 2 and d>2. For qubits wi…
Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared. In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an $N$-level quantum system. More precisely…
Monopoles play a center role in gauge theories and topological matter. There are two fundamental types of monopoles in physics: vector monopoles and tensor monopoles. Examples of vector monopoles include the Dirac monopole in 3D and Yang monopole in 5D, which have been extensively studied and observed in condensed matter or artificial systems. However, tensor monopoles are less studied, and their observation has not been reported. Here we experimentally construct a tunable spin-1 Hamiltonian to generate a tensor monopole and then measure its unique features with superconducting quantum circuit…
Unitary Schur Sampling of Qudits via Random SWAP Tests: Hunt for Antisymmetry
Duke Quantum Center, Duke University（杜克大学量子中心）；Department of Electrical and Computer Engineering, Duke University（杜克大学电气与计算机工程系）；Department of Physics, Duke University（杜克大学物理系）
本文提出仅用成对 SWAP 测试实现任意置换不变混合态的酉 Schur 采样，通过反复提取最大反对称子系统，将算法转化为 Young 图随机遍历，并应用于量子纯度放大以最优纯化态。
The CANDECOMP/PARAFAC (CP) decomposition is a generalization of the spectral decomposition of matrices to higher-order tensors. In this paper we use the CP decomposition to study unitary equivalence of higher order tensors and construct several invariants of local unitary equivalence for general higher order tensors. Based on this new method, we study the coefficient tensors of $3$-qubit states and obtain a necessary and sufficient criterion for local unitary equivalence of general tripartite states in terms of the CP decomposition. We also generalize this method to obtain some invariants of l…
Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials
