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Entangling four logical qubits beyond break-even in a nonlocal code
Yifan Hong,1, ∗Elijah Durso-Sabina,2David Hayes,2and Andrew Lucas1
1Department of Physics and Center for Theory of Quantum Matter,
University of Colorado, Boulder, CO 80309, USA
2Quantinuum, Broomfield, CO 80021, USA
(Dated: June 6, 2024)
Quantum error correction protects logical quantum information against environmental decoher-
ence by encoding logical qubits into entangled states of physical qubits. One of the most important
near-term challenges in building a scalable quantum computer is to reach the break-even point,
where logical quantum circuits on error-corrected qubits achieve higher fidelity than equivalent cir-
cuits on uncorrected physical qubits. Using Quantinuum’s H2 trapped-ion quantum processor, we
encode the GHZ state in four logical qubits with fidelity 99.5±0.15% ≤F≤99.7±0.1% (af-
ter postselecting on over 98% of outcomes). Using the same quantum processor, we can prepare
an uncorrected GHZ state on four physical qubits with fidelity 97.8±0.2% ≤F≤98.7±0.2%.
The logical qubits are encoded in a J25,4,3KTanner-transformed long-range-enhanced surface code.
Logical entangling gates are implemented using simple swap operations. Our results are a first step
towards realizing fault-tolerant quantum computation with logical qubits encoded in geometrically
nonlocal quantum low-density parity check codes.
Introduction.— It is widely believed that fault toler-
ance will be necessary for quantum computers to outper-
form classical computers on large-scale problems. De-
coherence, if left ignored, will eventually destroy entan-
glement and any quantum advantage in computation. A
critical milestone for the development of a quantum com-
puter will, therefore, be the protection of logical qubits
and the implementation of logical gates whose error rates
are below that of physical qubits and circuits which are
not error-corrected. When logical qubits are more robust
than physical qubits, we have reached break-even.
Since the discovery of topological quantum codes [1,2]
and their promise to preserve quantum information for
arbitrarily long times with a relatively high tolerance for
errors [3,4], there have been immense efforts to imple-
ment such codes in current architectures [5–9]. Unfor-
tunately, the resource requirements for large-scale, fault-
tolerant quantum computation using topological codes
with current hardware specifications may be enormous
[10].
More generally, it is known that these overheads can-
not be substantially improved in planar architectures
(codes) with only nearest-neighbor connectivity [11]. To
avoid this fundamental challenge, quantum low-density
parity-check (LDPC) codes [12–19] have been developed
to significantly reduce the resource overheads for error
correction [20,21]. To avoid the constraints of Ref. [11],
these codes will necessarily require long-range connectiv-
ity [22,23]. Platforms with such capabilities may be able
to significantly reduce the overhead for quantum error
correction, and a number of recent proposals have been
made for how to incorporate limited long-range connec-
tivity into hardware [24–27] to ameliorate the constraints
of spatial locality.
∗yifan.hong@colorado.edu
Thus far, we are not aware of any explicit implemen-
tation of a nonlocal LDPC code in any quantum hard-
ware. Other recent demonstrations of break-even opera-
tions we are aware of include the use of a small 7-qubit
Steane code [8,28] and the use of concatenated error
detection codes [9], neither of which achieve the high-
encoding rates of qLDPC codes. While such codes may
improve performance, to truly avoid large resource over-
heads in a scalable way, it is crucial to use a nonlocal code
which protects logical qubits in a more complex manner,
yet which is also amenable to quantum hardware.
This work provides a proof-of-principle demonstra-
tion that nonlocal codes can improve the performance
of present-day quantum computers. We have imple-
mented a J25,4,3Kquantum LDPC code (encoding 4 log-
ical qubits in 25 physical qubits, with the smallest logical
operator acting on 3 physical qubits) on Quantinuum’s
H2 trapped-ion quantum computer [29], which was con-
figured with 32 physical qubits at the time of the exper-
iments. This encoding enables us to prepare an error-
corrected, logical 4-qubit entangled GHZ state beyond
the break-even point, with mild postselection overhead,
for the first time. Using the existing hardware, it would
not have been possible to attempt such a demonstration,
even in principle, by using four surface code patches (with
36 total physical) to encode the logical qubits. This work
highlights the importance of choosing good hardware-
suited codes with efficient logical gates. The long-range-
enhanced surface code [26] underlying our methods will
be scalable to larger trapped-ion QCCD processors [30–
33] as well as to neutral atom arrays [8,34–37] for the
foreseeable future.
Quantum encoding.— The quantum LDPC code we
use is a Calderbank-Shor-Steane (CSS) stabilizer code
[38–40], which is defined by a stabilizer group of Pauli
strings that mutually commute and act trivially on log-
ical codewords. In a CSS code, these Pauli strings are
generated by products of strictly Xor Zoperators act-
arXiv:2406.02666v1 [quant-ph] 4 Jun 2024
2
ing on nqubits. Given the Pauli (anti)commutation re-
lations, it is helpful to organize data about the stabilizer
group in a pair of matrices HX,Z with F2={0,1}coeffi-
cients. Each row of the matrix HX, which in turn has n
columns, fixes one of the stabilizers of the code to be a
product of Pauli Xs on all qubits with a 1 in the corre-
sponding column; HZis defined similarly. To be a valid
code, we require Xand Zstabilizers to commute, which
means HXHT
Z= 0 (here multiplication/addition are mod
2).
A convenient way to build a quantum code is the hy-
pergraph product [13] of a classical linear code with itself.
For an input m×nclassical parity-check matrix H, the
resulting CSS stabilizer matrices take the form
HX=H⊗1n|1m⊗HT(1a)
HZ=1n⊗H|HT⊗1m.(1b)
If the classical input code has parameters [n, k, d] with
m=n−kparity checks (meaning that there are nclas-
sical bits encoding klogicals, with code distance d), then
so long as the classical code has no redundant parity
checks, the resulting quantum CSS hypergraph product
code has parameters Jn2+ (n−k)2, k2, dK. The geomet-
rical layout of the quantum code can be understood as
a Cartesian graph product between those of its classical
input codes. While the hypergraph product can produce
codes with constant rate (k∝n), the distance scaling
is sublinear (d≲√n). Nonetheless, this distance is pre-
served under standard syndrome extraction techniques
with circuit-level noise [41], a feature not supported by
other quantum codes such as the Steane code [40].
We now summarize the construction of our quantum
LDPC code, with a graphical depiction in Fig. 1. For
the classical input code, we first pick a [3,2,2] parity
code consisting of a single, global parity check on all
three physical bits. We then concatenate two of the
physical bits with a [2,1,2] repetition code to obtain a
[5,2,3] code. We then take the hypergraph product of
this [5,2,3] code with itself to obtain a J34,4,3KCSS
code. Finally, we perform a quantum Tanner transforma-
tion [19] to reduce the number of physical qubits from 34
to 25, while preserving both kand d. We note that the
quantum Tanner transformation does not preserve the
generic circuit-level distance property mentioned earlier;
however, we can still mitigate the effects of correlated er-
rors by choosing a clever pattern for syndrome extraction
like that for the rotated surface code [42]. The order of
CNOTs is carefully chosen so that all correlated ancilla
(hook) errors end up “against the grain” of the logical
operators, and so d= 3 errors on the ancilla qubits are
necessary to affect the codespace. This code is a minimal
example of a more general family of long-range-enhanced
surface codes [26], which possess the fewest nonlocal sta-
bilizers necessary to avoid the constraints of locality [11],
while adding O(1) logical qubits.
The standard protocol to prepare the GHZ state on 4
qubits is to initialize the qubits in |+000⟩and then apply
(a)
(b) (c)
FIG. 1. The code construction is illustrated. Circles and
red (blue) squares denote qubits and X(Z)-type stabilizers
respectively, with edges representing their connectivity. Ma-
genta lines denote long-range interactions. (a) A quantum
Tanner transformation reduces the number of physical qubits
from 34 to 25. (b) The support of the logical Xand Zop-
erators for logical qubits 1-4 are drawn with shaded boxes,
solid lines, dashed lines, and striped boxes respectively. (c)
The order of CNOT gates for distance-preserving syndrome
extraction is depicted for a selection of stabilizers.
CNOTs between the |+⟩qubit (control) and the rest (tar-
get); here Z|0⟩=|0⟩and X|+⟩=|+⟩. Since our code
is CSS and has a distance-preserving syndrome extrac-
tion circuit, we can perform a transversal initialization
into the logical |¯
0¯
0¯
0¯
0⟩state by initializing all physical
qubits in |0⟩⊗nand then measuring all X-type stabiliz-
ers. The stabilizer measurement can be performed by
initializing ancillas in the |+⟩state, performing CNOT
gates between the ancillas (control) and physical qubits
(target) according to the pattern in Fig. 1c, and finally
measuring all ancillas in the Xbasis. We then prepare
the third logical qubit in the |¯
+⟩state by measuring one
of its logical ¯
X3operators three times and postselecting
on the agreement of all three outcomes in order to mit-
igate the effects of readout error. After postselection, a
logical ¯
Z3is then applied (offline) if the agreed outcome is
−1. After preparing the logical |¯
0¯
0¯
+¯
0⟩state, we perform
the desired logical CNOT gates by permuting the data
qubits – a feature inherited from the [3,2,2] parity code.
This relabeling step can be done in software in our par-
ticular experiment. In particular, two combinations of
“fold-swaps” are sufficient: see Fig. 2. Finally, we per-
form a transversal measurement of all logical qubits in
either the Xor Zbasis by measuring all physical qubits
in that basis. This transversal measurement is then used
to reconstruct a final syndrome which is not subject to
3
(a)
(c) (d)
(b)
FIG. 2. The procedures for preparing the physical and logical
4-qubit GHZ states are illustrated. (a) The quantum circuit
which prepares the physical GHZ state. (b) The circuit which
prepares the logical GHZ state. (c-d) A combination of fold
swaps implements the above logical circuit while preserving
the stabilizer group.
circuit-level noise and can be used to correct errors. All
error correction is performed “in software” by tracking
the Pauli frame: the final logical measurement outcomes
are modified according to a decoding algorithm. We
use belief-propagation with ordered statistics decoding
(BP+OSD) [43,44], specifically the “minimum-sum” (10
iterations) and “combination sweep” (λ= 14) strategies
of BP and OS respectively, to infer physical errors based
on the syndrome measurement outcomes.
We now discuss the fault tolerance (or lack thereof)
of each of the aforementioned steps in our logical GHZ
protocol. As previously shown, our syndrome extraction
circuit is distance-preserving and so we do not need to
worry about ancilla hook errors. However, since we only
perform a single round of syndrome extraction, one may
worry about the effects of ancilla measurement errors.
For example, in a large surface code, a single flipped
check in the middle of the code may cause a matching
decoder to pair it to the boundary, resulting in a large
error string. Fortunately, for our distance three code, any
single flipped X-check can be caused by a single-qubit Z
error, and so we do not suffer from this type of error
propagation. For the logical ¯
X3measurement, a single
Zerror on a physical qubit prior to the measurements
can spoil the result, and so this step is not fault tolerant.
We note that a fault-tolerant procedure of measuring ¯
X3
can be done using generalized lattice surgery techniques
[45], though this would add considerable overhead (and
requires more physical qubits than were available in H2
at the time) and so we leave it to future work. The phys-
ical qubit permutations to implement the fold-swaps can
be achieved by a simple relabeling in software and so is
fault tolerant by definition. Lastly, the final measure-
GHZ experiment ¯zmismatch (%) ¯xmismatch (%)
Physical 1.3±0.2 0.9±0.1
Logical (No QEC) 5.7±0.5 2.9±0.3
Logical (with QEC) 0.3±0.1 0.2±0.1
TABLE 1. Fidelity of GHZ state preparation in experiment.
The second column displays the percentage of runs when the
four Zmeasurement outcomes disagreed. The third column
displays the percentage of runs when the product of the four
Xmeasurements was −1. Standard errors for the data are
also shown.
ment of all physical qubits is transversal and so is also
fault tolerant.
Experimental results.— The experiments described in
this work used the H2 trapped-ion processor described
in Ref. [29]. The H2 processor was configured with 32
trapped-ion qubits, all of which can be directly gated
with each other using ion-transport operations to pair
any two-qubits in a single potential well — a crucial fea-
ture for our J25,4,3Kcode. Physical-level entangling op-
erations are implemented using a Mølmer-Sørensen in-
teraction [46] driven by lasers directed to the four gating
regions. Likewise, physical level single-qubit gates, reset,
and measurement all take place in these same regions
using additional lasers systems. At the time of these ex-
periments, the typical error rates in H2 were as follows:
<2×10−3two-qubit gate error, ≈3×10−5single-qubit
gate error, and ≈2×10−3state preparation and mea-
surement (SPAM) error.
For the physical GHZ experiment, we run the circuit in
Fig. 2a for 10,000 shots, measuring all physical qubits in
the Zbasis for 5000 shots and in the Xbasis for the other
half. For the logical GHZ experiment, we run the circuit
in Fig. 2b for 5000 shots, with 2500 shots each dedicated
to logical ¯
Xand ¯
Zreadout. Uncertainties in the data are
computed using the standard error σ=pp(1 −p)/N,
where Nis the number of shots. For a perfect GHZ
state, we expect all Zmeasurement outcomes to agree
and the product of all Xmeasurement outcomes to have
even parity. The mismatch rates in the experimental
data from these theoretical predictions are tabulated in
Table 1. In addition to the error-corrected logical results,
we also give the uncorrected logical results which are ob-
tained by simply ignoring the syndrome values (i.e. no
feedback).
Since the GHZ state is a stabilizer state with gener-
ators {Z1Z2, Z2Z3, Z3Z4, X1X2X3X4}, we can compute
simple upper and lower bounds on the GHZ fidelity of
our experimental state ρ. Because |GHZ⟩is a pure state,
the fidelity [47] takes the form
FGHZ(ρ) = ⟨GHZ|ρ|GHZ⟩ ≡ ⟨ρ⟩GHZ .(2)
We now compute an upper bound for the experimental
fidelities. First, notice that the two projectors P±
zonto
whether the Zmeasurement outcomes agree (P+
z= (1+
Z1Z2)(1+Z2Z3)(1+Z3Z4)/8) or disagree (P−
z=1−P+
z)
4
form a projection-valued measure that splits the Hilbert
space into two orthogonal subspaces. Using the fact that
⟨P+
z⟩GHZ = 1, i.e. the GHZ state strictly lives in the P+
z
subspace, we can upper bound the fidelity (2) by
FGHZ(ρ)≤trP+
zρ,(3)
which corresponds to the second column in Table 1. For
the lower bound, observe that the GHZ state also strictly
resides in the subspace of P+
x≡(1+X1X2X3X4)/2 such
that its density matrix takes the form |GHZ⟩⟨GHZ|=
P+
zP+
x. We can then use the union bound from proba-
bility theory to upper bound the probabilities of being
orthogonal to GHZ, which provides a lower bound to the
fidelity (2):
FGHZ(ρ)≥1−trP−
zρ−trP−
xρ.(4)
The first average corresponds to the fraction of runs
where the Zoutcomes disagree and is given by the second
column in Table 1. The contribution from the XX X X
generator is given by the third column. For the physical
and (error-corrected) logical states ρ, ¯ρ, we hence obtain
fidelity bounds of
97.8±0.2% ≤FGHZ(ρ)≤98.7±0.2% (5a)
99.5±0.15% ≤FGHZ( ¯ρ)≤99.7±0.1% .(5b)
The statistical significance between the physical upper
bound and the logical lower bound is ≈3.6σ, assuming
statistically independent uncertainties.
Outlook.— We have prepared a logical GHZ state on
four qubits with a higher fidelity than its physical coun-
terpart. The key ingredient is a quantum LDPC code
whose long-range interactions enable (i) a compact en-
coding of logical qubits and (ii) efficient logical entan-
gling gates via qubit permutations. Having demonstrated
the ability to prepare specific quantum entangled logical
states beyond break-even, an immediate near-term ex-
perimental goal will be to use our encoding (or others)
to demonstrate break-even on larger circuits involving
multiple logical qubits, eventually including circuits with
non-Clifford gates.
Our work highlights an advantage of using quantum
LDPC codes in existing hardware over conventional topo-
logical codes like the surface and color codes. Of course,
a major feature of the topological codes is that they are
equipped with transversal implementations of the Clif-
ford group. A big challenge in the field is to construct
nonlocal quantum LDPC codes with hardware-efficient
logical gates; some progress using hypergraph product
codes has been previously made [48,49]. We hope that
this work stimulates further progress in this direction,
especially since it is already known that LDPC codes
can significantly reduce the overhead for fault-tolerant
quantum computation when functioning just as a mem-
ory block [45]. On a separate front, nonlocal codes de-
rived from concatenating quantum Hamming codes also
show promise for low-overhead fault tolerance [50,51].
Since these codes are not LDPC, they require more com-
plicated gadgets for fault tolerance. A worthwhile future
direction is to perform a detailed cost-benefit analysis be-
tween LDPC and concatenated architectures on different
hardware.
Acknowledgements.— This work was supported in part
by the Alfred P. Sloan Foundation under Grant FG-2020-
13795 (AL), by the Air Force Office of Scientific Re-
search under Grant FA9550-24-1-0120 (YH, AL), and by
the Office of Naval Research via Grant N00014-23-1-2533
(YH, AL). Additionally, we thank the hardware team at
Quantinuum for making these experiments possible.
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