Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing exponential separations between classical and quantum query complexity. While unitary designs originating from a group structure, such as the Clifford group, have proven to be incredibly useful for qubit systems, unfortunately, this is no longer true for qudits. In fact, the classification of finite-group representations rules out the existence of unitary 2-designs for arbitrary qudit dimensions. This severely limits the applicability of standard quantum information primitives when it comes to qudit systems. We overcome these limitations with a three-fold contribution. First, we introduce a general technique to construct families of weighted state t-designs in arbitrary qudit dimensions. These weighted state-designs generalize classical shadow tomography protocol from qubits to qudits. Second, we introduce a Clifford character RB that allows us to benchmark the qudit Clifford group in any dimension, including non-prime-power dimensions. And third, we establish bounds on the quantum circuit complexity of generating approximate unitary-designs from native gates in existing quantum hardware such as high-spin and cavity-QED qudits. Our work further highlights the analogy between spin and optical coherent states by proving that spin-GKP codewords form a state 2-design while spin coherent states do not; in direct analogy with the optical case. This work is structured as a pedagogical and self-contained introduction to unitary designs and their applications to qudit systems.
Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing exponential separations between classical and quantum query complexity. While unitary designs originating from a group structure, such as the Clifford group, have proven to be incredibly useful for qubit systems, unfortunately, this is no longer true for qudits. In fact, the classification of finite-group representations rules out the existence of unitary 2-designs for arbitrary qudit dimensions. This severely limits the applicability of standard quantum information primitives when it comes to qudit systems. We overcome these limitations with a three-fold contribution. First, we introduce a general technique to construct families of weighted state t-designs in arbitrary qudit dimensions. These weighted state-designs generalize classical shadow tomography protocol from qubits to qudits. Second, we introduce a Clifford character RB that allows us to benchmark the qudit Clifford group in any dimension, including non-prime-power dimensions. And third, we establish bounds on the quantum circuit complexity of generating approximate unitary-designs from native gates in existing quantum hardware such as high-spin and cavity-QED qudits. Our work further highlights the analogy between spin and optical coherent states by proving that spin-GKP codewords form a state 2-design while spin coherent states do not; in direct analogy with the optical case. This work is structured as a pedagogical and self-contained introduction to unitary designs and their applications to qudit systems.
Unitary t-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary t-designs with $t\ensuremath{\ge}3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension 4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, the set of stabilizer states forms a 3-design. In addition, our study is helpful in studying higher moments of the Clifford group, which are useful in many research areas ranging from quantum information science to signal processing. Furthermore, we reveal a surprising connection between unitary 3-designs and the physics of discrete phase spaces and thereby offer a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group, which is of intrinsic interest in studying quantum computation.
A unitary t-design is a powerful tool in quantum information science and fundamental physics.Despite its usefulness, only approximate implementations were known for general t.In this paper, we provide quantum circuits that generate exact unitary t-designs for any t on an arbitrary number of qubits.Our construction is inductive and is of practical use in small systems.We then introduce a tth-order generalization of randomized benchmarking (t-RB) as an application of exact 2t-designs.We particularly study the 2-RB in detail and show that it reveals self-adjointness of quantum noise, a metric related to the feasibility of quantum error correction (QEC).We numerically demonstrate that the 2-RB in one-and two-qubit systems is feasible, and experimentally characterize background noise of a superconducting qubit by the 2-RB.It is shown from the experiment that interactions with adjacent qubits induce the noise that may result in an obstacle toward a realization of QEC.
Qudits are the natural generalization of qubits, that is, the basic unit of information in quantum computation. This generalization allows for quantum states to be represented with the same cyclic nature present in qubits, while also permitting the basic unit of information to be stored in d-level systems. In this document, we are concerned with studying universality over qudits on arbitrary dimension d. The question of single-qudit universality of a set of operators can be reduced to studying the adjoint representation of the set and determining whether or not its generated group is finite. Thus, we set out to find universal sets of operators in quantum computation over qudits, for any possible value of d. Moreover, we focus on finding “Clifford + T-like” universal sets of operators in arbitrary dimensions, in the hopes of maintaining a strong analog to the qubits case and its most studied universal set. We accomplish this goal by studying how the Clifford group behaves on arbitrary dimensions, and how this behavior relates to the typical Clifford group in prime dimensions. Furthermore, we find that the Clifford group on arbitrary dimension splits as the typical Clifford group on each of the coprime factors of d, which leads to a fundamental difference between the study of universality on prime power dimensions and arbitrary dimensions containing co-prime factors. Then, we focus on studying the structure of universal sets in dimension d, for any d containing at least two co-prime factors. Thus, after studying which T-like gates drive the single-qudit Clifford group to universality on prime power dimensions; that is, guaranteeing the irreducibility of the adjoint representation and an infinite generated group, we combine these two approaches to obtain “Clifford + T-like” universal sets on all possible dimensions.
Unitary designs are essential tools in several quantum information protocols. Similarly to other design concepts, unitary designs are mainly used to facilitate averaging over a relevant space, in this case, the unitary group U ( d ) . While it is known that exact unitary t -designs exist for any degree t and dimension d , the most appealing type of designs, group designs (in which the elements of the design form a group), can provide at most 3 -designs. Moreover, even group 2 -designs can exist only in limited dimensions. In this paper, we present novel construction methods for creating exact generalized group designs based on the representation theory of the unitary group and its finite subgroups that overcome the 4 -design-barrier of unitary group designs. Furthermore, a construction is presented for creating generalized group 2 -designs in arbitrary dimensions.
Unitary designs provide finite-moment approximations to Haar-random unitaries, with wide-ranging applications across physics and quantum information, from scrambling and black-hole dynamics to foundational primitives in quantum algorithms. Strong unitary designs capture a more demanding operational notion of approximation, requiring indistinguishability from Haar randomness even for quantum algorithms that may access a unitary not only in the forward direction, but also through its inverse, transpose, and complex conjugate. Motivated by the physical requirement that scrambling arise within the system itself, Schuster, Ma, Lombardi, Brandão, and Huang (arXiv:2509.26310) left open whether strong unitary designs can be generated in logarithmic depth using only the system qubits. For every fixed design order k and measurable-error tolerance, we construct strong approximate unitary k-designs in optimal Θ(logn) all-to-all circuit depth using only the n original system qubits. Our new ingredient is a logarithmic-depth Pauli-mixing bound for the perfect-matching ensemble, whose layers pair the qubits uniformly at random and apply independent random two-qubit gates. This bound controls the mixed forward-reverse two-query case, which we combine with existing design and gluing results to obtain strong unitary designs of arbitrary fixed order.
Unitary T-designs play an important role in quantum information, with diverse applications in quantum algorithms, benchmarking, tomography, and communication. Until now, the most efficient construction of unitary T-designs for n-qudit systems has been via random local quantum circuits, which have been shown to converge to approximate T-designs in the diamond norm using O(T5+o(1)n2) quantum gates. In this work, we provide a new construction of T-designs via random matrix theory using O~(T2n2) quantum gates. Our construction leverages two key ideas. First, in the spirit of central limit theorems, we approximate the Gaussian Unitary Ensemble (GUE) by an i.i.d. sum of random Hermitian matrices. Second, we show that the product of just two exponentiated GUE matrices is already approximately Haar random. Thus, multiplying two exponentiated sums over rather simple random matrices yields a unitary T-design, via Hamiltonian simulation. A central feature of our proof is a new connection between the polynomial method in quantum query complexity and the large-dimension (N) expansion in random matrix theory. In particular, we show that the polynomial method provides exponentially improved bounds on the high moments of certain random matrix ensembles, without requiring intricate Weingarten calculations. In doing so, we define and solve a new type of moment problem on the unit circle, asking whether a finite number of equally weighted points, corresponding to eigenvalues of unitary matrices, can reproduce a given set of moments.