Entanglement Purification with Quantum LDPC Codes and Iterative Decoding

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Entanglement Purification with Quantum LDPC
Codes and Iterative Decoding
Narayanan Rengaswamy1, Nithin Raveendran1, Ankur Raina2, and Bane Vasić1
1Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA
2Department of Electrical Engineering and Computer Sciences, Indian Institute of Science Education and Re-
search, Bhopal, Madhya Pradesh 462066, India
Recent constructions of quantum low-density parity-check (QLDPC) codes
provide optimal scaling of the number of logical qubits and the minimum dis-
tance in terms of the code length, thereby opening the door to fault-tolerant
quantum systems with minimal resource overhead. However, the hardware
path from nearest-neighbor-connection-based topological codes to long-range-
interaction-demanding QLDPC codes is likely a challenging one. Given the
practical difficulty in building a monolithic architecture for quantum systems,
such as computers, based on optimal QLDPC codes, it is worth considering
adistributed implementation of such codes over a network of interconnected
medium-sized quantum processors. In such a setting, all syndrome measure-
ments and logical operations must be performed through the use of high-fidelity
shared entangled states between the processing nodes. Since probabilistic
many-to-1 distillation schemes for purifying entanglement are inefficient, we
investigate quantum error correction based entanglement purification in this
work. Specifically, we employ QLDPC codes to distill GHZ states, as the
resulting high-fidelity logical GHZ states can interact directly with the code
used to perform distributed quantum computing (DQC), e.g. for fault-tolerant
Steane syndrome extraction. This protocol is applicable beyond the applica-
tion of DQC since entanglement distribution and purification is a quintessen-
tial task of any quantum network. We use the min-sum algorithm (MSA)
based iterative decoder with a sequential schedule for distilling 3-qubit GHZ
states using a rate 0.118 family of lifted product QLDPC codes and obtain
an input threshold of ≈0.7974 under i.i.d. single-qubit depolarizing noise.
This represents the best threshold for a yield of 0.118 for any GHZ purifi-
cation protocol. Our results apply to larger size GHZ states as well, where
we extend our technical result about a measurement property of 3-qubit GHZ
states to construct a scalable GHZ purification protocol. Our software is avail-
able at: https://github.com/nrenga/ghz_distillation_qec/tree/main/
qldpc-ghz_protocol_II and https://zenodo.org/record/8284903.
Narayanan Rengaswamy: narayananr@arizona.edu, A shorter version of this work was presented at the 2023 Inter-
national Symposium on Topics in Coding (https://ieeexplore.ieee.org/abstract/document/10273456).
Nithin Raveendran: nithin@arizona.edu
Ankur Raina: ankur@iiserb.ac.in
Bane Vasić: vasic@ece.arizona.edu
Accepted in Quantum 2024-01-11, click title to verify. Published under CC-BY 4.0. 1
arXiv:2210.14143v2 [quant-ph] 16 Jan 2024
1 Introduction
ADVANCES in quantum technologies are happening at a breathtaking pace and these
will lead to exciting applications in quantum computing, networking, sensing, secu-
rity, and more. Quantum networking is a common theme in all these applications, such as
for employing quantum key distribution to enhance digital security, for connecting quan-
tum sensors together to enable a quadratic gain in sensing precision, and for distributing
quantum computation among multiple quantum processors to relax the burden of building
enormous monolithic quantum computers. This work is primarily motivated by the latter
role of quantum networking. Indeed, for fault-tolerant quantum computing (FTQC), the
best codes for scalability are the recently proposed constructions of quantum low-density
parity-check (QLDPC) codes [1,2,3,4,5,6]. They provide optimal scaling of the code
parameters, i.e., the number of logical qubits and the minimum distance, with respect to
the length of the code, and thereby form promising candidates for FTQC with minimal
resource overhead. While topological codes such as the surface code are also QLDPC
codes, they encode only a fixed number of logical qubits even with diverging code size and
have suboptimal scaling of the minimum distance. However, they just require nearest-
neighbor connections to build in hardware, whereas these optimal QLDPC codes require
many long-range connections. Even though the LDPC property means that each stabi-
lizer check involves only a fixed number of qubits and similarly each qubit is only involved
in a fixed number of checks, both irrespective of the code size, there are a large number
of connections between checks and qubits that are non-local geometrically [7]. Thus, it
becomes very challenging to build such codes in practice for several technologies such as
superconducting qubits.
Given such practical constraints, it becomes very relevant and interesting to explore
Distributed Quantum Computing (DQC): a distributed realization of these QLDPC codes
where multiple interconnected medium-sized quantum processors each store a subset of
qubits and coordinate processing through the means of a classical compute node. Naturally,
this means that all the logical operations and syndrome measurements on the coded qubits
are now non-local, i.e., must involve multiple nodes. Such an architecture was explored by
Nickerson et al. [8] even a decade ago, but in the context of the surface code. The solution
to perform non-local operations is to share high-fidelity entangled Bell and GHZ states
among the nodes, perform local gates between code qubits and these ancillary entangled
qubits, and pool the classical measurement results across nodes to assess the state of the
qubits. For example, in the case of the surface code with each node possessing only one
code qubit, each 4-qubit syndrome measurement will involve one CNOT per node between
the code qubit and one of the 4qubits of an ancillary GHZ state shared between the
nodes; this is followed by a single-qubit Pauli measurement on the ancillary qubit and
classical communication of the result with other nodes. The authors proposed to produce
high-fidelity 4-qubit GHZ states by generating Bell pairs between pairs of nodes and then
“fusing” them to form the GHZ state. The process involved multiple rounds of simple
probabilistic purification of the entangled state, which is in general inefficient since the
number of consumed Bell pairs can be very large (and uncertain). While their hand-
designed purification schemes have been extended by algorithmic procedures recently [9,
10], the approach still suffers from this inefficiency arising from the heralded nature of the
protocol. We will discuss comparisons to past work on GHZ purification after we present
our results in the next section.
Our goal in this paper is to investigate a principled and systematic procedure to purify
(or distill) GHZ states using quantum error correcting codes (QECCs). If one can use the
Accepted in Quantum 2024-01-11, click title to verify. Published under CC-BY 4.0. 2
same QLDPC codes that DQC will employ for FTQC (“compute code”) to also store logical
GHZ ancillary states, then these can potentially be directly interacted with the compute
code for performing fault-tolerant (e.g., Steane) syndrome extraction and measurement-
based methods for logical operations. Thus, it is very pertinent to develop a scalable GHZ
purification protocol using these optimal QLDPC codes (“purification code”). While DQC
is a key motivation, such a protocol serves a much wider purpose, since entanglement
generation, distribution and purification form the cornerstone of quantum networking. For
efficient and scalable quantum networks, one must necessarily deploy quantum repeaters
whose primary function is to help entangle different subsets of parties in the network. In
the long-run, third generation quantum repeaters will use quantum error correction for
entanglement purification [11]. Such repeater nodes, and other nodes of the network that
are not quantum computers, will still need to possess a fault-tolerant quantum memory
to generate and store (shares of) high-fidelity entangled states. Therefore, if the compute
nodes will deploy QLDPC codes, then QLDPC purification codes could potentially unify
the functioning of different parts of the network and enable seamless integration.
2 Main Results and Discussion
Entanglement purification is a well-studied problem in quantum information, where one
typically starts with ncopies of a noisy Bell pair, or a general mixed state, and distills k
Bell pairs of higher fidelity [12]. Several teams of researchers have worked on this problem,
and the contributions range from fundamental limits [12,13,14,15,16,17] to simple and
practical protocols [9,13,18,19]. Of course, if one can distill Bell pairs, then these can
be “fused” in sequence to entangle multiple parties, but direct distillation of an entangled
resource between all parties can be more efficient [20]. Some purification schemes involve
two-way communications between the involved parties while others only need one-way
communication. We focus on one-way schemes in this paper. The connection between
one-way entanglement purification protocols (1-EPPs) and QECCs was established by
Bennett et al. in 1996 [13]. They showed that any QECC can be converted into a 1-EPP
(and vice-versa). This framework enables systematic n-to-kprotocols where the rate and
average output fidelity are directly a function of the QECC rate and decoding performance,
respectively. Since the recently constructed QLDPC codes have asymptotically constant
rate and linear distance scaling with code size [1,2,3,4,5,6], our work paves the way for
high-rate high-fidelity entanglement distillation.
2.1 Purifying Bell Pairs with QLDPC Codes
In 2007, Wilde et al. [18] showed that any classical convolutional code can be used to
distill Bell pairs via their entanglement assisted 1-EPP scheme. In the development of
this scheme, they mention a potentially different method to use a QECC for performing
1-EPP [18, Section II-D] (without entanglement assistance), compared to the protocol by
Bennett et al. Initially, Alice generates nperfect Bell pairs locally, marks one qubit of each
pair as ‘A’ and the other as ‘B’, and measures the stabilizers of her chosen [[n, k]] code
on qubits ‘A’. Due to the “transpose” property of Bell states, this simultaneously projects
qubits ‘B’ onto an equivalent code (see Appendix A.4). Then, she performs a local Pauli
operation on qubits ‘A’ to fix her obtained syndrome, shares her code stabilizers and
syndrome with Bob through a noiseless classical channel, and sends qubits ‘B’ to Bob
over a noisy Pauli channel. Using the transpose property, Bob measures the appropriate
code stabilizers on qubits ‘B’, and corrects channel errors by combining his syndrome with
Accepted in Quantum 2024-01-11, click title to verify. Published under CC-BY 4.0. 3
10−210−1
10−6
10−5
10−4
10−3
10−2
10−1
100
Depolarizing Probability
Logical Error Rate
[[n= 544, k = 80, d = 12]]
[[n= 714, k = 100, d = 16]]
[[n= 1020, k = 136, d = 20]]
0.100 0.101 0.102 0.103 0.104 0.105 0.106 0.107 0.108
0.65
0.7
0.75
0.8
0.85
0.9
Depolarizing Probability
Logical Error Rate
[[n= 544, k = 80, d = 12]]
[[n= 714, k = 100, d = 16]]
[[n= 1020, k = 136, d = 20]]
Figure 1: (top) The performance of a family of lifted product QLDPC codes with asymptotic rate
0.118 using the sequential schedule of the min-sum algorithm (MSA) based decoder. Each data point
is obtained by counting 100 logical errors. (bottom) The threshold is about 10.6-10.7%. These results
apply to Bell pair purification, up to a rescaling of the depolarizing probabilities.
Accepted in Quantum 2024-01-11, click title to verify. Published under CC-BY 4.0. 4
Alice’s syndrome. Finally, Alice and Bob decode their respective qubits, i.e., invert the
encoding unitary (see Appendix A.2), to convert the klogical Bell pairs into kphysical
Bell pairs. Since the code corrects some errors, on average the output Bell pairs are of
higher fidelity than the initial nnoisy ones.
As our first contribution, we elucidate this protocol for general stabilizer codes [21,22]
through the lens of the stabilizer formalism [23], using the 5-qubit perfect code [21,24]
as an example. This approach clarifies many details of the protocol, especially from an
error correction standpoint, and helps adapt it to different scenarios. For the performance
of the protocol, note that any error on Alice’s qubits can be mapped into an equivalent
error on Bob’s qubits using the transpose property, in effect increasing the error rate on
Bob’s qubits. Therefore, since only Bob corrects errors in this protocol, the failure rate of
the protocol is the same as the logical error rate (LER) of the code on the depolarizing
channel, with an effective channel error rate that accounts for errors on Alice’s qubits as
well as Bob’s qubits (as long as they do not amount to a Bell state stabilizer together).
If errors only happen on Bob’s qubits, then the failure rate of the protocol is identical to
the logical error rate of the code. For all simulations in this work, we consider a rate 0.118
family of lifted product (LP118) QLDPC codes decoded using the sequential schedule of
the min-sum algorithm (MSA) based iterative decoder with normalization factor 0.8and
maximum number of iterations set to 100 [25,26]. The LER of this code-decoder pair is
shown in Fig. 1, where we see that the threshold is about 10.6-10.7%. Since the fidelity
is one minus the depolarizing probability, this translates to an input fidelity threshold
of about 89.3-89.4%. Also, even with n= 544, the LER is ≈10−6at depolarizing rate
10−2. Again, note that these curves can be interpreted as the performance of Bell pair
purification when only Bob’s qubits are affected by errors.
2.2 New Protocols to Purify GHZ States with QLDPC Codes
Protocol I: Given these insights, we proceed to investigate the purification of GHZ states.
As in the Wilde et al. protocol, we consider only local operations and one-way classical
communications (LOCC), and assume that these are noiseless. The key technical insight
necessary to construct the protocol is the GHZ-equivalent of the transpose property of
Bell pairs. Given ncopies of the GHZ state, whose three subsystems are marked ‘A’,
‘B’ and ‘C’, we find that applying a matrix on qubits ‘A’ is equivalent to applying a
“stretched” version of the matrix on qubits ‘B’ and ‘C’ together (see Lemma 3). We call this
mapping to the stretched version of the matrix the GHZ-map, and prove that it is an algebra
homomorphism [27], i.e., linear, multiplicative, and hence projector-preserving. Recollect
from the Bell pair purification setting that we are interested in measuring stabilizers on
qubits ‘A’ and understanding their effect on the remaining qubits. Using the properties of
the GHZ-map, we show that it suffices to consider only the simple case of a single stabilizer.
With this great simplification, we prove that imposing a given [[n, k, d]] stabilizer code on
qubits ‘A’ simultaneously imposes a certain [[2n, k, d′]] stabilizer code jointly on qubits ‘B’
and ‘C’. By performing diagonal Clifford operations on qubits ‘C’, which commutes with
any operations on the other qubits, one can vary the distance d′of the induced ‘BC’ code.
Then, we use this core technical result to devise a natural protocol that purifies GHZ states
using any stabilizer code (“Protocol I”, see Fig. 2and Algorithm 2).
We perform simulations on the [[5,1,3]] perfect code and compare the protocol failure
rate to the LER of the code on the depolarizing channel, both using a maximum-likelihood
decoder. In terms of error exponents, we show that it is always better for Bob to perform
a local diagonal Clifford operation on Charlie’s qubits, rather than Alice doing the same.
We support the empirical observation with an analytical argument on the induced BC code
Accepted in Quantum 2024-01-11, click title to verify. Published under CC-BY 4.0. 5
摘要：

EntanglementPurificationwithQuantumLDPCCodesandIterativeDecodingNarayananRengaswamy1,NithinRaveendran1,AnkurRaina2,andBaneVasić11DepartmentofElectricalandComputerEngineering,UniversityofArizona,Tucson,Arizona85721,USA2DepartmentofElectricalEngineeringandComputerSciences,IndianInstituteofScienceEduca...

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