bosonic exchange statistics, their combination
yexhibits fermionic exchange statistics. The
mutual and exchange statistics of the anyons,
conventionally summarized in the modularS
andTmatrices, fully characterize theℤ 2 topo-
logical order ( 29 ).
Preparation of logical qubit states
Distinct topologically ordered ground states
are locally indistinguishable, making them
attractive logical qubits thanks to this immunity
to local perturbations. The lattice of Fig. 1A has
only one ground state under Eq. 1, but as
shown in Fig. 4A, we used different boundary
conditions in which the toric code admits a
ground-state degeneracy, as proposed for the
surface code ( 5 , 6 , 36 ). We introduced logical
operatorsZLandXL, which span across the lat-
tice and commute with the Hamiltonian but
anticommute with each other.
We generalized the state preparation circuit
of Fig. 1B to create the logical states 0jiL and
jiþL, whereZLji (^0) L ¼þ 10 jiL andXLjiþL¼
þ 1 jiþL, on both 5-by-5 (distance-5) and 3-by-3
(distance-3) arrays. The 0jiLandjiþL prep-
arations are closely related, connected by a
logical Hadamard gate. We then used the
logical operators, which are simply products of
single-qubit gates, to realize 1jiL ¼XLji (^0) Land
j i¼L ZLjiþL. Details on state preparation
and logical operations are provided in ( 24 ),
section I.
The logical states are resilient to local errors,
whichwedemonstratedwithlogicalmeasure-
ment with error correction (Fig. 4B). Following
surface code proposals, we performed a logical
measurement by projectively measuring all the
qubits inZorXbasis (forZLorXL, respectively).
Naively evaluating the parity of the logical
operator is vulnerable to errors on any qubit
along the operator, but errors can be detected
by also evaluating the local parities (Asor
Bp) from the individual qubit measurements.
By construction, we expected the local par-
ities to be +1, so any–1 values indicate near-
by errors. We found a minimal set of qubits
to flip in order to recover +1 parities before
evaluating the logical operator. This correc-
tion decreases the logical error substantially.
Averaging logical preparation and measure-
ment error overXLandZLeigenstates, without
correction, we observed 0.17 for distance-5
and 0.090 for distance-3, whereas with cor-
rection, we observed 0.030 for both, which is
lower than the average physical qubit prep-
aration and measurement error, 0.034. This
is a simplified form of error correction com-
pared with the repetitive stabilizer measure-
ments of surface code proposals, in which
parity changes are matched together over
space and time.
The logical subspace also admits arbitrary
superposition statesaji (^0) Lþbji (^1) L, which we
realized with state injection, encoding a single
physical qubit state into the logical qubit. For
5-by-5 state injection, we prepared the cen-
tral qubit inaj iþ 0 bji 1 and then created a
GHZ-like stateðÞaIþbXL 0
25 using three
CZ layers. The toric code preparation circuit
maps 0ji^25 →ji (^0) LandXLji 0
25 →ji (^1) L, giving
aj iþ (^0) L bji (^1) L. For example, the states depicted
in Fig. 4A are logicalTstatesjiTL ¼ðji (^0) L þ
eip=^4 j iÞ (^1) L =
ffiffiffi
2
p
, which is of interest for non-
Clifford operations. We characterized injected
states using logical tomography. Measurement
ofZLandXLwas straightforward and robust.
We measuredYLby performing another logical
gate,X
1 = 2
L ¼ðÞIiXL=
ffiffiffi
2
p
, decomposed into
five CZ layers, and then measuringZL.Weplot
the resultant Bloch vectors for 128 injected
states across the Bloch sphere in Fig. 4C. By
measuring these nonlocal order parameters,
we illuminate the logical degree of freedom
that was invisible to the local parity measure-
ments of Fig. 4A. In contrast to surface code
proposals, our state-preparation methods are
not fault tolerant, although in principle, states
prepared in this way can be further purified
through distillation ( 7 , 8 , 37 , 38 ).
Last, we investigated decoherence ofZLand
XLeigenstates by plotting logical error versus
wait timet(Fig. 4D). We continued our focus
on measurement error correction through
comparison of the raw and corrected data.
Although distance-5 has substantially worse
raw error, after correction it is modestly better
than distance-3 for 1jiL. However,jiþLdecays
much more quickly than 1jiL, owing to its
sensitivity toZerrors (dephasing). We dy-
namically decoupled the qubits from low-
frequency noise with a simple sequence that
executed anXgate on each qubit att/4 and
3 t/4, which broughtjiþLerror slightly below
ji (^1) Lerror, with distance-3 remaining slightly
lower-error compared with that of distance-
- 1jiLand 0jiLare not appreciably affected by
this dynamical decoupling [( 24 ), section V].
Overall, the logical error increases linearly at
0.06 per microsecond. For active error cor-
rection with the surface code, we expected a
few percent logical error per cycle at thresh-
old ( 9 ). Typical cycle durations are hundreds
of nanoseconds ( 39 ), in which the logical state
suffers the decoherence studied here as well as
1240 3 DECEMBER 2021•VOL 374 ISSUE 6572 science.orgSCIENCE
Fig. 3. Extracting braiding statistics by using Ramsey interferometry.(A) Visualizing braiding with a
toric code excited statej i¼E U 1 jiG[excitationse(red) andm(yellow), experimentally measured parities].
We applied additionalXgates (U 2 ,U 3 , andU 4 ) to visualize braiding anearound them.(B) Quantum
circuits with unitaryUand an eigenstatejif. (Left) Direct application. (Right) Extracting the phaseqby using
an auxiliary qubit (green). (C) Illustration of Ramsey interferometry for the case of braiding aneandm
(statejif) by using an operatorU. We visualized the superposition of two paths, with the braid operationU
controlled by an auxiliary qubit injiþ.(D) Extracting the mutual statistics foreandm. (Left) Initial
excited eigenstate [similar to that in (A)]. We implemented controlled-XXXXwith an auxiliary control qubit
(green) starting injiþ. (Right) Parity measurements after controlled-XXXX.(E) Extracting the fermion
exchange statistics, analogous to (D). We created two pairs ofy(neighboringeandm) and implemented
controlled-XXYYZZto measure the exchange phase. (F) Measured mutual and exchange phases, with braiding
diagrams. Phases are from tomography on the auxiliary qubit, 18,000 total repetitions per compiled
instance. Standard error was estimated with jackknife resampling over instances.
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