PRX QUANTUM 6, 030333 (2025)
Emergent Unitary Designs for Encoded Qubits from Coherent Errors and
Syndrome Measurements
Zihan Cheng ,
1,*
Eric Huang ,
2
Vedika Khemani,
3
Michael J. Gullans ,
2
and Matteo Ippoliti
1
1
Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA
2
Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park,
Maryland 20742, USA
3
Department of Physics, Stanford University, Stanford, California 94305, USA
(Received 26 April 2025; revised 10 July 2025; accepted 24 July 2025; published 22 August 2025)
Unitary k-designs are distributions of unitary gates that match the Haar distribution up to its kth
statistical moment. They are a crucial resource for randomized quantum protocols. However, their imple-
mentation on encoded logical qubits is nontrivial due to the need for magic gates, which can require a
large resource overhead. In this work, we propose an efficient approach to generate unitary designs for
encoded qubits in surface codes by applying local unitary rotations (“coherent errors”) on the physical
qubits followed by syndrome measurement and error correction. We prove that, under some conditions
on the coherent errors (notably including all single-qubit unitaries) and on the error-correcting code, this
process induces a unitary transformation of the logical subspace. We numerically show that the ensem-
ble of logical unitaries (indexed by the random syndrome outcomes) converges to a unitary design in the
thermodynamic limit, provided that the density or strength of coherent errors is above a finite threshold.
This “unitary design” phase transition coincides with the code’s coherent error threshold under optimal
decoding. Furthermore, we propose a classical algorithm to simulate the protocol based on a “staircase”
implementation of the surface code encoder and decoder circuits. This enables a mapping to a (1 + 1)-
dimensional monitored circuit, where we observe an entanglement phase transition (and thus a classical
complexity phase transition of the decoding algorithm) coinciding with the aforementioned unitary design
phase transition. Our results provide a practical way to realize unitary designs on encoded qubits, with
applications including quantum state tomography and benchmarking in error-correcting codes.
DOI: 10.1103/bnld-2chd
I. INTRODUCTION
Quantum error correction is crucial to the realization
of scalable quantum computations with real, noisy hard-
ware. At the same time, encoded quantum information
tends to be harder to manipulate. Quantum error-correcting
codes have some restricted set of logical operations that
can be implemented “transversally,” i.e., by separately
rotating individual physical qubits—this is desirable for
fault tolerance, as transversal operations do not propa-
gate errors. But, these transversal logical operations are
not universal for quantum computing [1]. For surface
codes and other codes of practical interest, “magic” (i.e.,
*
Contact author: zihan_cheng@utexas.edu
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license. Fur-
ther distribution of this work must maintain attribution to the
author(s) and the published article’s title, journal citation, and
DOI.
non-Clifford) gates are among those that cannot be imple-
mented transversally. Strategies to overcome this issue,
such as magic state distillation, come with some over-
head in terms of auxiliary qubits or circuit size [2–5].
This motivates the search for alternative strategies that can
implement generic unitary operations on encoded qubits
without additional resources.
In this work we address this issue within the context
of random unitary gates, which are a key component
of various quantum information processing protocols like
randomized benchmarking, randomized measurements for
state and process learning, cryptography, random cir-
cuit sampling, quantum simulation, etc. Our strategy is
based on the application of transversal physical opera-
tions (which may be seen as “coherent errors,” though
applied intentionally and known to the experimentalist)
followed by syndrome extraction and error correction, as
sketched in Fig. 1(a). The Born-rule randomness inher-
ent in quantum measurements automatically gives rise to a
random distribution of operations on the encoded informa-
tion; remarkably, under some criteria that we characterize,
2691-3399/25/6(3)/030333(22) 030333-1 Published by the American Physical Society
