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

By Christoph Dankert, Richard Cleve, Joseph Emerson, Etera R. Livine. 2006

697 citations recorded by the source corpus.

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.

Subfields: Hardware and Architecture, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a8549c910f7255d98814581e42c4738e9b341494

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Citations in the corpus

Black holes as mirrors: Quantum information in random subsystems (2007, methodology): “If we replace the integral over all unitary transformations by an average over the elements of the ǫ-approximate unitary 2-design K constructed in [ 6 ], then eq. (3.5) is modified to become”
Detecting crosstalk errors in quantum information processors (2019, neutral): “noise because they twirl it [14, 20, 7].”
Emergent Quantum State Designs from Individual Many-Body Wave Functions (2021, background): “Therefore, this approach to generating random states is extremely hardware efficient and could lead to various useful applications in quantum information science [14,15,17,18,32,33].”
Unitary designs from statistical mechanics in random quantum circuits. (2019, background): “But for general k, very few constructions of exact k-designs are known, with the exception of k = 2 and k = 3 [17–21].”
Noise tailoring for scalable quantum computation via randomized compiling (2015, background): “Moreover, by the definition of a unitary 1-design [21], the above sum is independent of the choice of T and so is a Pauli channel for any unitary 1-design.”
Emergent statistical mechanics of entanglement in random unitary circuits (2018, background): “[50] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine.”
Local random quantum circuits are approximate polynomial-designs: numerical results (2012, neutral): “[27] C.”
Barren plateaus in variational quantum computing (2024, neutral): “In fact, equation (7) precludes any type of non-exponentially suppressed features in the landscape.”
Linear growth of quantum circuit complexity (2021, neutral): “[42] A t-design is a probability distribution, over unitaries, whose first t moments equal the Haar measure’s moments [17, 23].”
Stringy effects in scrambling (2014, background): “This timescale made another appearance in the work of Hayden and Preskill [8], who made the connection between the quantum chaotic process of fast scrambling [9–16] and the Planckian boost time t∗.”
Barren plateaus in quantum neural network training landscapes (2018, background): “Whereas a unitary 2-design will exhibit the correct variance [43, 46], a unitary 1-design will exhibit the correct average value, but not necessarily the variance.”