Unitary 2-design Research Articles

Emergence of noise-induced barren plateaus in arbitrary layered noise models

In variational quantum algorithms the parameters of a parameterized quantum circuit are optimized in order to minimize a cost function that encodes the solution of the problem. The barren plateau phenomenon manifests as an exponentially vanishing dependence of the cost function with respect to the variational parameters, and thus hampers the optimization process. We discuss how, and in which sense, the phenomenon of noise-induced barren plateaus emerges in parameterized quantum circuits with a layered noise model. Previous results have shown the existence of noise-induced barren plateaus in the presence of local Pauli noise (Wang et al 2021 Nat. Commun. 12 6961). We extend these results analytically to arbitrary completely-positive trace preserving maps in two cases: (1) when a parameter-shift rule holds, (2) when the parameterized quantum circuit at each layer forms a unitary 2-design. The second example shows how highly expressive unitaries give rise not only to standard barren plateaus (McClean et al 2018 Nat. Commun. 9 4812), but also to noise-induced ones. In the second part of the paper, we study numerically the emergence of noise-induced barren plateaus in QAOA circuits focusing on the case of MaxCut problems on d-regular graphs and amplitude damping noise.

Approximate unitary 3-designs from transvection Markov chains

Unitary k-designs are probabilistic ensembles of unitary matrices whose first k statistical moments match that of the full unitary group endowed with the Haar measure. In prior work, we showed that the automorphism group of classical \({\mathbb {Z}}_4\)-linear Kerdock codes maps to a unitary 2-design, which established a new classical-quantum connection via graph states. In this paper, we construct a Markov process that mixes this Kerdock 2-design with symplectic transvections, and show that this process produces an \(\epsilon \)-approximate unitary 3-design. We construct a graph whose vertices are Pauli matrices, and two vertices are connected by directed edges if and only if they commute. A unitary ensemble that is transitive on vertices, edges, and non-edges of this Pauli graph is an exact 3-design, and the stationary distribution of our process possesses this property. With respect to the symmetries of Kerdock codes, the Pauli graph has two types of edges; the Kerdock 2-design mixes edges of the same type, and the transvections mix the types. More precisely, on m qubits, the process samples \(O(\log (N^5/\epsilon ))\) random transvections, where \(N = 2^m\), followed by a random Kerdock 2-design element and a random Pauli matrix. Hence, the simplicity of the protocol might make it attractive for several applications. From a hardware perspective, 2-qubit transvections exactly map to the Mølmer–Sørensen gates that form the native 2-qubit operations for trapped-ion quantum computers. Thus, it might be possible to extend our work to construct an approximate 3-design that only involves such 2-qubit transvections.

Character Randomized Benchmarking for Non-Multiplicity-Free Groups With Applications to Subspace, Leakage, and Matchgate Randomized Benchmarking

Randomized benchmarking (RB) is a powerful method for determining the error rate of experimental quantum gates. Traditional RB, however, is restricted to gatesets, such as the Clifford group, that form a unitary 2-design. The recently introduced character RB can benchmark more general gates using techniques from representation theory; up to now, however, this method has only been applied to "multiplicity-free" groups, a mathematical restriction on these groups. In this paper, we extend the original character RB derivation to explicitly treat non-multiplicity-free groups, and derive several applications. First, we derive a rigorous version of the recently introduced subspace RB, which seeks to characterize a set of one- and two-qubit gates that are symmetric under SWAP. Second, we develop a new leakage RB protocol that applies to more general groups of gates. Finally, we derive a scalable RB protocol for the matchgate group, a group that like the Clifford group is non-universal but becomes universal with the addition of one additional gate. This example provides one of the few examples of a scalable non-Clifford RB protocol. In all three cases, compared to existing theories, our method requires similar resources, but either provides a more accurate estimate of gate fidelity, or applies to a more general group of gates. In conclusion, we discuss the potential, and challenges, of using non-multiplicity-free character RB to develop new classes of scalable RB protocols and methods of characterizing specific gates.

Trace-2 excluded subsets of special linear groups over finite fields and mutually unbiased maximally entangled bases

Mutually unbiased maximally entangled bases (MUMEBs) in bipartite systems have a close relation with unitary 2-design and have attracted much attention in recent years. In the paper, we construct MUMEBs in $$\mathbb {C}^q\otimes \mathbb {C}^q$$ with q a power of an odd prime number. For this purpose, we introduce the notation of trace-2 excluded subset of the special linear group $$SL(2,\mathbb {F}_q)$$ over the finite field $$\mathbb {F}_q$$ and establish a relation between a trace-2 excluded subset and a set of MUMEBs in $$\mathbb {C}^q\otimes \mathbb {C}^q$$ . Under this relation, we prove that $$M(q,q)\ge \dfrac{q^2-1}{2}$$ by constructing trace-2 excluded subsets in $$SL(2,\mathbb {F}_q)$$ , which highly raises the lower bound of M(q, q) given in Liu et al. (Quantum Inf Process 16(6):159, 2017) and Xu (Quantum Inf. Process 16(3):65, 2017).

Minimax quantum state estimation under Bregman divergence

We investigate minimax estimators for quantum state tomography under general Bregman divergences. First, generalizing the work of Koyama et al. [Entropy 19, 618 (2017)] for relative entropy, we find that given any estimator for a quantum state, there always exists a sequence of Bayes estimators that asymptotically perform at least as well as the given estimator, on any state. Second, we show that there always exists a sequence of priors for which the corresponding sequence of Bayes estimators is asymptotically minimax (i.e. it minimizes the worst-case risk). Third, by re-formulating Holevo's theorem for the covariant state estimation problem in terms of estimators, we find that there exists a covariant measurement that is, in fact, minimax (i.e. it minimizes the worst-case risk). Moreover, we find that a measurement that is covariant only under a unitary 2-design is also minimax. Lastly, in an attempt to understand the problem of finding minimax measurements for general state estimation, we study the qubit case in detail and find that every spherical 2-design is a minimax measurement.

Average fidelity estimation of twirled noisy quantum channel using unitary 2t-design**Project supported by the National Natural Science Foundation of China (Grant Nos. 61372076 and 61701375), the 111 Project, China (Grant No. B08038), the Key Research and Development Plan of Shannxi Province, China (Grant No. BBD24017290001), and the Foundation of Science and Technology on Communication Networks Laboratory, China (Grant No. KX172600031).

We propose a method to estimate the average fidelity using the unitary 2t-design of a twirled noisy channel, which is suitable for large-scale quantum circuits. Compared with the unitary 2-design in randomized benchmarking, the unitary 2t-design for the twirling of noisy channels is more flexible in construction and can provide more information. In addition, we prove that the proposed method provides an efficient and reliable estimation of the average fidelity in benchmarking multistage quantum gates and estimating the weakly gate- and time-dependent noise. For time-dependent noise, we provide a scheme of moment superoperator to analyze the noise in different experiments. In particular, we give a lower bound on the average fidelity of a channel with imperfect implementation of benchmarking and state preparation and measurement errors (SPAM).

Recovering Quantum Gates from Few Average Gate Fidelities.

Characterizing quantum processes is a key task in the development of quantum technologies, especially at the noisy intermediate scale of today's devices. One method for characterizing processes is randomized benchmarking, which is robust against state preparation and measurement errors and can be used to benchmark Clifford gates. Compressed sensing techniques achieve full tomography of quantum channels essentially at optimal resource efficiency. In this Letter, we show that the favorable features of both approaches can be combined. For characterizing multiqubit unitary gates, we provide a rigorously guaranteed and practical reconstruction method that works with an essentially optimal number of average gate fidelities measured with respect to random Clifford unitaries. Moreover, for general unital quantum channels, we provide an explicit expansion into a unitary 2-design, allowing for a practical and guaranteed reconstruction also in that case. As a side result, we obtain a new statistical interpretation of the unitarity-a figure of merit characterizing the coherence of a process.

Multiqubit Clifford groups are unitary 3-designs

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 $t\geq3$ 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 to 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 to studying quantum computation.

Unitary 2-designs from random X- and Z-diagonal unitaries

Unitary 2-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary 2-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-Z basis and Pauli-X basis. We also provide a converse about the number of repetitions needed to achieve unitary 2-designs. These results imply that the process after ℓ repetitions achieves a Θ(d−ℓ)-approximate unitary 2-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary 2-designs. Although a more efficient implementation of unitary 2-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary 2-design after switching interactions only a few times.

Estimating the fidelity of T gates using standard interleaved randomized benchmarking

Randomized benchmarking is an important protocol for robustly characterizing the error rates of quantum gates. The technique is typically applied to the Clifford gates since they form a group that satisfies a convenient technical condition of forming a unitary 2-design, in addition to having a tight connection to fault-tolerant quantum computing and an efficient classical simulation. In order to achieve universal quantum computing one must add at least one additional gate such as the T gate (also known as the π/8 gate). Here we propose and analyze a simple variation of the standard interleaved RB protocol that can accurately estimate the average fidelity of the T gate while retaining the many advantages of a unitary 2-design and the fidelity guarantees that such a design delivers, as well as the efficient classical simulation property of the Clifford group. Our work complements prior methods that have succeeded in estimating T gate fidelities, but only by relaxing the 2-design constraint and using a more complicated data analysis.

Near-linear constructions of exact unitary 2-designs

A unitary 2-design can be viewed as a quantum analogue of a 2-universal hash function: it is indistinguishable from a truly random unitary by any procedure that queries it twice. We show that exact unitary 2-designs on n qubits can be implemented by quantum circuits consisting of Oe(n) elementary gates in logarithmic depth. This is essentially a quadratic improvement in size (and in width times depth) over all previous implementations that are exact or approximate (for sufficiently strong approximations).

Experimental estimation of average fidelity of a Clifford gate on a 7-qubit quantum processor.

One of the major experimental achievements in the past decades is the ability to control quantum systems to high levels of precision. To quantify the level of control we need to characterize the dynamical evolution. Full characterization via quantum process tomography is impractical and often unnecessary. For most practical purposes, it is enough to estimate more general quantities such as the average fidelity. Here we use a unitary 2-design and twirling protocol for efficiently estimating the average fidelity of Clifford gates, to certify a 7-qubit entangling gate in a nuclear magnetic resonance quantum processor. Compared with more than 10^{8} experiments required by full process tomography, we conducted 1656 experiments to satisfy a statistical confidence level of 99%. The average fidelity of this Clifford gate in experiment is 55.1%, and rises to at least 87.5% if the signal's decay due to decoherence is taken into account. The entire protocol of certifying Clifford gates is efficient and scalable, and can easily be extended to any general quantum information processor with minor modifications.

Exact and approximate unitary 2-designs and their application to fidelity estimation

We develop the concept of a unitary t-design as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group U(2^n) on n qubits. In particular, sets of unitaries forming 2-designs have wide applicability to quantum information protocols. We devise an O(n)-size in-place circuit construction for an approximate unitary 2-design. We then show that this can be used to construct an efficient protocol for experimentally characterizing the fidelity of a quantum process on n qubits with quantum circuits of size O(n) without requiring any ancilla qubits, thereby improving upon previous approaches.

Nonmalleable encryption of quantum information

We introduce the notion of nonmalleability of a quantum state encryption scheme (in dimension d): in addition to the requirement that an adversary cannot learn information about the state, here we demand that no controlled modification of the encrypted state can be effected. We show that such a scheme is equivalent to a unitary 2-design [Dankert, et al., e-print arXiv:quant-ph/0606161], as opposed to normal encryption which is a unitary 1-design. Our other main results include a new proof of the lower bound of (d2−1)2+1 on the number of unitaries in a 2-design [Gross, et al., J. Math. Phys. 48, 052104 (2007)], which lends itself to a generalization to approximate 2-design. Furthermore, while in prime power dimension there is a unitary 2-design with ≤d5 elements, we show that there are always approximate 2-designs with O(ϵ−2d4 log d) elements.