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June 1, 2022 1
Quantum Speedup for Higher-Order
Unconstrained Binary Optimization and MIMO
Maximum Likelihood Detection
Masaya Norimoto, Student Member, IEEE, Ryuhei Mori, Non Member, IEEE, and
Naoki Ishikawa, Senior Member, IEEE
Abstract—In this paper, we propose a quantum algorithm that supports a real-valued higher-order unconstrained binary optimization
(HUBO) problem. This algorithm is based on the Grover adaptive search that originally supported HUBO with integer coefficients. Next,
as an application example, we formulate multiple-input multiple-output maximum likelihood detection as a HUBO problem with
real-valued coefficients, where we use the Gray-coded bit-to-symbol mapping specified in the 5G standard. The proposed approach
allows us to construct a specific quantum circuit for the detection problem and to analyze specific numbers of required qubits and
quantum gates, whereas other conventional studies have assumed that such a circuit is feasible as a quantum oracle. To further
accelerate the convergence, we also derive a probability distribution of the objective function value and determine a unique threshold to
sample better states for the quantum algorithm. Assuming a future fault-tolerant quantum computer, we demonstrate that the proposed
algorithm is capable of reducing the query complexity in the classical domain and providing a quadratic speedup in the quantum
domain.
Index Terms—Grover adaptive search (GAS), quadratic unconstrained binary optimization (QUBO), higher-order unconstrained binary
optimization (HUBO), multiple-input multiple-output (MIMO), maximum-likelihood detection (MLD).
F
1 INTRODUCTION
MARCONI invented a practical long-range wireless sys-
tem in 1895. Since then, driven by its intense demand,
wireless communication has continued to become more so-
phisticated as if there were no limits. The limit of communi-
cation throughput is known as the Shannon capacity, which
is constrained by the bandwidth, the signal-to-noise ratio
(SNR), and the numbers of transmit and receive antennas for
multiple-input multiple-output (MIMO) scenarios. Clearly,
there are physical limits on bandwidth, SNR, and the num-
ber of antennas. The forward error correction techniques
such as the low-density parity-check code (LDPC) and po-
lar code can achieve near-capacity performance efficiently,
but under a certain energy constraint, their performance is
constrained by semiconductor miniaturization limits. Semi-
conductor companies may abandon their pursuit of Moore’s
law in the 2020s [1, 2]. Marconi mentioned in 1932 that
it is dangerous to put limits on wireless. However, wireless
communication will reach its physical limits in the near
future.
After the eventual end of Moore’s law, from a long-
term perspective, we must rely on a different computing
paradigm, and quantum computing in particular is be-
lieved to be promising. Since it is impossible to simulate
a quantum computer in an efficient manner on a classi-
cal computer, quantum computers offer an essential speed
•M. Norimoto and N. Ishikawa are with the Faculty of Engineering,
Yokohama National University, 240-8501 Kanagawa, Japan.
E-mail: ishikawa-naoki-fr@ynu.ac.jp
•R. Mori is with the Department of Mathematical and Computing Sciences,
School of Computing, Tokyo Institute of Technology, 152-8500 Tokyo,
Japan.
advantage over classical computers [3]. Specifically, Shor ’s
algorithm [4] factors an n-bit integer with the complexity
O(n2log nlog log n), while the best classical algorithm re-
quires exp(Θ(n1/3log2/3n)) operations [3],1which is an
exponential speedup. Grover’s algorithm [6] finds a specific
element from a database of unsorted Nelements with
the query complexity O(√N), while the classic exhaus-
tive search requires O(N)evaluations, which is a quadratic
speedup.
Grover’s algorithm has been extended to support binary
optimization problems. The pioneering algorithm, Grover
adaptive search (GAS) [7], requires a complex quantum cir-
cuit to evaluate an objective function. To solve this issue,
in [8, 9], Gilliam et al. used a quantum dictionary and
allowed for the representation of an arbitrary polynomial
function, including quadratic and higher-order terms. It
was described in [9] that an example quadratic uncon-
strained binary optimization (QUBO) problem with integer
coefficients was solved on a real-world quantum computer
equipped with 32 qubits. Unlike other approaches such as
the quantum annealing [10] and the quantum approximate
optimization algorithm [11], the GAS proposed by Gilliam
et al. is innovative in that it supports a higher-order uncon-
strained binary optimization (HUBO) problem with integer
coefficients, which cannot be solved efficiently with state-
of-the-art mathematical programming solvers on a classical
computer, such as CPLEX and Gurobi.
In designing wireless systems, the trade-off between
performance and complexity is in general a source of
1. O(·)denotes the big-Onotation, while Θ(·)denotes the big-Θ
notation [5].
arXiv:2205.15478v1 [eess.SP] 31 May 2022
June 1, 2022 2
concern for engineers and researchers. For example, low-
complexity MIMO detectors and polar decoders inevitably
involve the penalty of lower performance, and complexity is
sacrificed to achieve optimal performance. In this situation,
the speedup capability of quantum algorithms has inspired
those who dream of striking the fundamental trade-off and
achieving the optimal performance with reduced complex-
ity. A pioneering attempt in wireless communications was
provided in [12]. In [12], Botsinis et al. demonstrated the
potential of quantum search algorithms to reduce the com-
plexity involved in maximum likelihood detection (MLD).
Specifically, they used the Grover-type algorithms, such
as Boyer–Brassard–Høyer–Tapp (BBHT) [13] and the D ¨
urr–
Høyer (DH) searches [14], for performing MLD of data
symbols on a quantum computer [15]. Following [12], a
number of important studies have shown promising results
[15–22]. However, in those studies, it was assumed that
an ideal quantum circuit to evaluate the objective function
is feasible as a quantum oracle, which will be detailed in
Section 2. For more information on quantum optimization
in wireless communications, a comprehensive survey can
be found in [23, 24].
Against this background, we propose a quantum al-
gorithm that supports a HUBO problem with real-valued
coefficients. Then, as a first step toward breaking the trade-
off between performance and complexity, we formulate the
MIMO MLD as a real-valued HUBO problem and verify the
capability of quadratic speedup. The major contributions of
this paper are organized as follows.
1) While the conventional GAS [9] supports HUBO
with integer coefficients, we modify the quantum
algorithm to handle real-valued coefficients. This
allows us to solve a HUBO problem even if the
objective function contains real-valued coefficients,
at the cost of one more query in the classical domain
(CD).
2) As an application example, we formulate the ob-
jective function of MIMO MLD as a real-valued
HUBO problem. This formulation is not a straight-
forward task because the objective function contains
complex-valued random variables and a Frobenius
norm calculation. This new formulation allows us
to analyze specific numbers of qubits and quantum
gates required in the constructed quantum circuits,
which has been overlooked in conventional studies.
3) We clarify the probability distribution of the ob-
jective function value and determine the threshold
used inside GAS more efficiently. Then, we demon-
strate that the proposed threshold further acceler-
ates the convergence of GAS to the optimal solution.
It is important to note that quantum circuits are sensitive
to noise [25], and industrial applications require decades of
effort and challenge. The noise induces quantum errors, and
quantum error-correcting codes must be used to perform
reliable arithmetic on a quantum computer. For example,
if we use the surface code with code distance 27, which is
one of the quantum error-correcting codes, a logical qubit
requires 1568 physical qubits to correct errors [26]. This
indicates that even a simple quantum circuit with fewer
qubits, e.g., as in Fig. 1, may require many more physical
TABLE 1
List of important mathematical symbols
BBinary numbers
RReal numbers
CComplex numbers
ZIntegers
Nt∈ZNumber of transmit antennas
Nr∈ZNumber of receive antennas
Lc∈ZModulation order (constellation size)
σ2∈RNoise variance
γ∈RSignal-to-noise ratio
E(·)∈ZObjective function
n∈ZNumber of variables =transmission rate
m∈ZNumber of qubits required to encode E(·)
i∈ZIndex of GAS iterations
y, yi∈ZThreshold that is adaptively updated by GAS
L, Li∈ZNumber of Grover operators
P∈RProbability that controls the proposed threshold
b,bi∈BnBinary variables, or data bits
s∈CNt×1Data symbols, each symbol is denoted by st
r∈CNr×1Received symbols, each symbol is denoted by ru
Hc∈CNr×NtChannel coefficients, hut
v∈CNr×1Additive white Gaussian noise, vu
qubits. Since this limitation is out of the scope of our
contributions, we assume the realization of a future fault-
tolerant quantum computer.
The remainder of this paper is organized as follows.
Section 2 is a review of important related works, while in
Section 3, we introduce the conventional GAS and its mod-
ification to support real-valued coefficients. In Section 4,
a method to solve MIMO MLD on a quantum computer
is proposed, and algebraic and numerical evaluations are
given in Section 5. Finally, in Section 6, we conclude this
paper. Italicized symbols represent scalar values, and bold
symbols represent vectors and matrices. Table 1 summarizes
a list of important mathematical symbols used in this paper.
2 RE LATED WOR KS
Quantum computation has the potential to break through
the fundamental trade-off between performance and com-
plexity. Hence, it has been applied to multi-user detection
[12, 15–17, 19, 27, 28], multiple symbol differential detection
[18], channel coding [29, 30], wireless routing [20, 21], in-
door localization [22], intelligent reflecting surfaces [31, 32],
and codeword optimization problem [33]. In this section,
we introduce important related works targeting detection
problems in wireless communications.
2.1 Multi-User Detection Using DH Algorithm [15]
Botsinis et al. proposed a novel method of applying the
DH algorithm to multi-user detection [15], which is a de-
tection problem for multi-user scenarios. The original DH
algorithm [14] is terminated if the sum of the number of
Grover iterations becomes greater than or equal to 22.5√N,
where Ndenotes the search space size. By contrast, Botsinis
et al. modified the algorithm to terminate early for an
arbitrary number of queries smaller than 22.5√N. Addi-
tionally, the modified algorithm calculates the output of a
low-complexity detector, such as the zero-forcing (ZF) or
minimum mean square error (MMSE) detector, and exploits
the output as an initial value to sample better states. Both
June 1, 2022 3
contributions are innovative in that they accelerate the
quantum algorithm more for a specific problem in wireless
communications.
The objective function presented in [15] involves a Frobe-
nius norm of complex-valued variables. However, the quan-
tum circuit that evaluates the norm is idealized as an oracle,
and no specific construction method is considered. Unlike
in [15], we consider specific quantum circuits and analyze
their hardware and query complexities, which is the missing
piece in the literature.
2.2 MIMO MLD Using Quantum Annealing [27]
Kim et al. formulated MIMO MLD as a QUBO problem
and solved it using quantum annealing, the D-Wave 2000Q
quantum annealer [27]. Specifically, binary phase-shift key-
ing (BPSK) and quadrature phase-shift keying (QPSK) sym-
bols are represented as first-order functions with respect to
information bits, while gray-coded 16 quadrature amplitude
modulation (QAM) symbols are represented as second-
order functions. Since the objective function of MLD con-
tains the squared norm, it may result in a higher-order
function such as fourth, eighth, or higher, which is not
supported by quantum annealing. To solve this problem,
Kim et al. used first-order functions that represent higher-
order modulation, such as 16-QAM or 64-QAM, without the
Gray coding. Then, the objective function contains first- and
second-order terms only. To achieve performance equivalent
to that of the Gray-coded case, the projection between before
and after Gray coding is used in CD. That is, encoding at the
transmitter and decoding at the receiver require additional
steps.
Unlike in the above study [27] targeting quantum an-
nealing, we directly handle the Gray-coded data symbols
specified in the 5G standard owing to the proposed real-
valued GAS that supports higher-order terms. Our ap-
proach is capable of supporting any signal modulation,
such as star-QAM and constellation shaping schemes, as
long as data symbols can be represented as a function of
information bits.
2.3 MIMO MLD Using DH Algorithm [28]
Mondal et al. proposed a method to solve MIMO MLD
using the DH algorithm [28]. Specifically, to improve the
success probability of the algorithm, the uniform selection
of the number of Grover operators, L, was modified to a
random value from the Gamma distribution, leading to a
better selection of L. Here, the Gamma distribution depends
on a scale parameter, and the scale parameter depends on
the exact number of solutions to be marked. Since the exact
number of solutions varies dynamically depending on the
threshold, the quantum counting algorithm [34] is crucial,
as stated in [28]. Additionally, the concept of reducing the
search space was verified.
As with [15], a specific construction method for a quan-
tum circuit is not considered in [28]. Herein, we determine
a threshold in accordance with the distribution of objective
function values, which is known in advance.
3 GROVER ADAPTIVE SEARCH (GAS)
GAS [9] supports binary optimization problems with inte-
ger coefficients, including QUBO and HUBO problems. It
requires nqubits for nbinary variables b∈Bnand mqubits
for encoding the objective function value E(b)∈Z, result-
ing in a circuit equipped with n+mqubits. The classic ex-
haustive search requires O(2n)queries, while GAS requires
O(√2n)queries, which provides a quadratic speedup. GAS
obtains a global minimum solution by amplifying the states
in which the objective function value E(b)is smaller than
the current threshold yi∈Z. Here, yiis a temporal min-
imum and iis an iteration count in CD. We measure the
quantum states and update the threshold, which is repeated
until a termination condition is satisfied.
Before running GAS, it is not a straightforward task to
determine an appropriate number of qubits m. The objective
function value is expressed as two’s complement. Let the
objective function value or its coefficient be an integer k.
Then, mmust satisfy [9]
−2m−1≤k < 2m−1.(1)
As the threshold yiis updated in each iteration of GAS,
the calculated value may become E(b)−yi, which results
in a smaller minimum value or a larger maximum value.
Thus, it is necessary to set a sufficient mthat might handle
Emax +Emin without overflow, where Emax and Emin are
the maximum and minimum of E(b), respectively.
3.1 Conventional GAS for Integer QUBO [9]
We review a specific construction method for the quantum
circuit used in GAS. First, a state preparation operator Ay
is constructed, in which an n-qubit input register is trans-
formed into the equal superposition of all states and an m-
qubit input register is used to represent the corresponding
value E(b)−y. Taking the binary variables bas a binary
number and converting it to a decimal number b, the state
should be [9]
Ay|0in|0im=1
√2n
2n−1
X
b=0 |bin|E(b)−yim.(2)
This operator Aycan be composed of the Hadamard gates
H, controlled unitary operators UG(θ), and the inverse
quantum Fourier transform (IQFT). Let kbe a constant
term in the objective function. The non-controlled unitary
operator UG(θ)is defined such that [9]
UG(θ)H⊗m|0im=1
√2m
2m−1
X
l=0
ejlθ |lim,(3)
where we have θ= 2πk/2m. That is, it is constructed by
UG(θ) = R(2m−1θ)⊗R(2m−2θ)⊗ · ·· ⊗ R(20θ)(4)
and the phase gate
R(θ) = 1 0
0ejθ .(5)
Here, phase advance represents integer addition and phase
delay represents subtraction. Following (3), IQFT yields only
one state that represents the original integer value of k.
June 1, 2022 4
IQFT
Fig. 1. Quantum circuit corresponding to E(b) = 1 + b0−2b1b2.
(a) L= 0. (b) L= 1. (c) L= 2.
Fig. 2. Output probabilities of the circuit shown in Fig. 1
The interaction between a binary variable and a coeffi-
cient can be represented by a controlled qubit. Similarly, the
interaction between binary variables can be represented by
controlled qubits on a register |bin. As exemplified in Fig. 1,
the constant term +1 corresponds to UG(π/4), the term
+1b0corresponds to controlled UG(π/4), and the term
−2b1b2corresponds to controlled UG(−2π/4). Likewise,
higher-order terms, such as third or fourth order, can be
represented by increasing the number of controlled qubits.
In the classic Grover search [6], an oracle operator O
identifies the states of interest and inverts the phases of
these states. Only the inverted states are amplified by the
Grover operator. The operator Ayabove calculates the
values E(b)−yfor all 2nstates in parallel. Here, states
that are better than the current threshold y,i.e., states that
satisfy E(b)−y < 0, should be marked to find the minimum
solution. Since the calculated values are represented by the
two’s complement, we can identify the negative states by
focusing only on the beginning of the mqubits, and Ocan
be constructed by applying the Z-gate only to that qubit.
Let the Grover diffusion operator be D[6]. The Grover
operator is finally constructed by G=AyDAH
yO, and
we evaluate GLAy|0in+m, which will maximize the ampli-
tudes of the states of interest. The ideal Lthat successfully
maximizes the amplitude is given by [35]
Lopt =$π
4sN
Ns%,(6)
where Ndenotes the search space size, 2n, and Nsdenotes
the number of solutions.
Algorithm 1 Conventional GAS designed for integer coeffi-
cients [9].
Input: E:Bn→Z, λ = 8/7
Output: b
1: Uniformly sample b0∈Bnand set y0=E(b0).
2: Set k= 1 and i= 0.
3: repeat
4: Randomly select the rotation count Lifrom the set
{0,1, ..., dk−1e}.
5: Evaluate GLiAyi|0in+m, and obtain band y.
{Grover search}
6: if y < yithen
7: bi+1 =b, yi+1 =y, and k= 1.{Improvement
found}
8: else
9: bi+1 =bi, yi+1 =yi,and k= min {λk, √2n}.
{No Improvement}
10: end if
11: i=i+ 1.
12: until a termination condition is met.
From (6), the query complexity of GAS can be derived
as O(√2n)[9] in the quantum domain (QD). Since Ns,
the number of states better than the current threshold, is
unknown in advance, Lis typically drawn from a uniform
distribution ranging from 0to a specific value that increases
by a factor of λ= 8/7at each iteration. GAS is terminated
if the sum of the Grover operators is greater than 22.5√2n,
which is the same as the conventional DH algorithm. In the
Qiskit implementation, the number of times no improve-
ment is observed is also considered as one of termination
conditions. Overall, GAS is summarized in Algorithm 1.
As a specific example, Fig. 1 shows a quantum cir-
cuit of GAS that tries to minimize the objective function
E(b) = 1 + b1−2b2b3. The Hadamard gate at the beginning
initializes the qubits and creates an equal superposition of
all the possible states, 000000 to 111111. The black circle
in Fig. 1 indicates a control qubit. The unitary operator
UG(θ)is applied if all the associated control qubits are 1.
Here, if the control qubit is in a superposition state, the gate
creates a quantum entanglement state, which plays a key
role in GAS. Fig. 2 shows the probability that each state
is measured, where the number of Grover operators was
varied from L= 0 to 2. The comma-separated text in this
figure shows nand mqubits, and the latter is converted
to a decimal number. As shown in Fig. 2, when L= 0,
23= 8 different states were observed with equal probabil-
ity, and the corresponding values of the objective function
were correctly calculated, demonstrating the potential of
quantum computation. When L= 1 and 2, only the state
of interest b= [0 1 1], which yields E(b) = −1<0,
was successfully amplified by the Grover operator. In this
manner, GAS amplifies the states that are better than the
current threshold and finds a binary solution that minimizes
the objective function.
3.2 Handling of Real-Valued Coefficients [9]
Polynomials may contain real-valued coefficients. For deal-
ing with real-valued coefficients, Gilliam et al. proposed the
following two methods [9].
June 1, 2022 5
3.2.1 Integer Approximation
Multiplying the objective function by a positive constant
does not affect the minimization process. A real-valued
coefficient can be approximated by multiplying a large
number and rounding down to an integer. Specifically, real
coefficients are approximated as fractions with a common
denominator, the denominator is multiplied to the objective
function, and the numerators become approximated integer
coefficients. As can be inferred from (1), the drawback is
that the number of required qubits mincreases as the value
range of the objective function expands. If mis kept small,
this approximation becomes less accurate.
3.2.2 Direct Encoding
In this method, an integer kin θ= 2πk/2mof (3) is replaced
with a real-valued coefficient a∈R. Then, the output
probability indicates multiple integers, which is known as
the Fej´
er distribution. Specifically, the state UFej´er(θ)|0im
after applying IQFT to UG(θ)H⊗m|0imis given by [9]2
UFej´er(θ)|0im=
2m−1
X
l=0 hg(θ),g(2πl/2m)i |li,(7)
where we have θ= 2πa/2mand g(θ) =
[1, ejθ ,··· , ej(2m−1)θ]/√2m. The number of qubits m
must satisfy [9]
−2m−1≤a < 2m−1.(8)
In this distribution, the probabilities of two integers close to
a given real number aare greater than the other probabili-
ties. For example, if m= 3 qubits and a=−2.5, from (7),
−2and −3are observed with equal probability. If a=−2.3,
−2is observed more frequently than −3.
3.3 Proposed GAS for Real-Valued HUBO
As previously reviewed in Section 3.2, in the innovative
study [9], Gilliam et al. proposed two methods for handling
real-valued coefficients, but did not specifically investigate
how GAS behaves in the case of direct encoding. In such
a case, in our evaluation, GAS samples a wrong value of
the objective function, which obeys the Fej´
er distribution.
For example, if the objective function value is −2.5, we may
observe an integer value less than or equal to −3. A value
lower than the actual value is updated as the minimum and
set as a new threshold y. Then, no states satisfy E(b)−y < 0,
and one of all states is randomly sampled. As a result, GAS
will not be able to obtain an optimal solution.
A possible solution here is that we ignore yevaluated
in QD. Instead, we use breturned by GAS and calculate
a correct objective function value y=E(b)in CD. Since
the quantum circuit using the direct encoding amplifies the
states of interest with high probability, with this simple
modification, GAS obtains an optimal solution correctly.
Overall, the above procedure is summarized in Algorithm 2.
We have two major drawbacks. First, it increases query
complexity in CD, although the asymptotic order remains
the same. Second, the probability amplification may not be
sufficient, which is illustrated in Fig. 4.
2. This definition differs from [9], but is essentially identical.
Algorithm 2 Proposed GAS designed for real-valued coeffi-
cients.
Input: E:Bn→R, λ = 8/7
Output: b
1: Uniformly sample b0∈Bnand set y0=E(b0).{This
step will be improved in Section 4.3}
2: Set k= 1 and i= 0.
3: repeat
4: Randomly select the rotation count Lifrom the set
{0,1, ..., dk−1e}.
5: Evaluate GLiAyi|0in+m, and obtain b.
6: Evaluate y=E(b)in CD. {This is the additional
step}
7: if y < yithen
8: bi+1 =b, yi+1 =y, and k= 1.
9: else
10: bi+1 =bi, yi+1 =yi,and k= min {λk, √2n}.
11: end if
12: i=i+ 1.
13: until a termination condition is met.
:
IQFT
Fig. 3. Quantum circuit corresponding to E(b) = 1 + b0−1.8b1b2b3.
(a) L= 0. (b) L= 1. (c) L= 2. (d) L= 3.
Fig. 4. Output probabilities of the circuit shown in Fig. 3, where only the
top 16 states were shown.
As a specific example, Fig. 3 shows a quantum circuit
corresponding to the objective function E(b) = 1 + b0−
1.8b1b2b3, where we set n= 4 and m= 3. Since we used
the direct encoding method, −1.8b1b2b3was represented as
UG(−1.8π/4), and it was associated with three qubits |b1i,
|b2i, and |b3i. Additionally, Fig. 4 shows the output proba-
bilities of Fig. 3, where only the top 16 states were shown
for simplicity. The state of interest here is b= [0 1 1 1] and
June 1, 2022 6
E(b) = 1 −1.8 = −0.8<0. As shown in Fig. 4, the states
(0111,−1) and (0111,−2) were amplified as the number
of Grover operators Lincreased. The bad state (0111,−2)
was observed with a lower probability than (0111,−1).
The wrong state (1111,−1) was also observed with a low
probability. This is the reason why the correction of the
objective function value is required for real-valued GAS, as
summarized in Algorithm 2.
3.4 Evaluation Metrics
In the literature, a quantum circuit has been evaluated by
the number of qubits, gates, and its depth, while a quantum
algorithm has been evaluated by query complexity.
3.4.1 Number of Qubits, Gates and Depth
The size of the quantum circuit determines its feasibility.
As the number of qubits and gates in a quantum circuit
increases, more advanced quantum computation becomes
possible. At the same time, however, it becomes more
susceptible to noise and more difficult to implement in
hardware. In our evaluations, the number of required qubits
is represented as n+m, and the number of quantum gates
is derived as a function with repect to nand m.
3.4.2 Query Complexity [15]
To investigate query complexity, we count how many times
the objective function is queried. Specifically, the query com-
plexity in the classical domain (CD) is the number of times
the objective function is evaluated, i.e., iin Algorithm 1. By
contrast, the query complexity in the quantum domain (QD)
is the number of times the Grover operator Gis applied, i.e.,
L0+L1+· ·· +Liin Algorithm 1.
4 QUA NTUM SPEEDUP FOR MIMO MLD
Conventional studies on quantum-assisted wireless com-
munications have not considered a specific construction
method of quantum circuit. In many cases, the circuit to
calculate an objective function has been idealized as a black-
box quantum oracle. In this section, we formulate the MIMO
MLD as a new real-valued HUBO problem, which can be
represented by a quantum circuit, as described in Section 3.
We also analyze the probability distribution of the objective
value for enabling further speedup.
4.1 System Model
We consider a MIMO communication scenario with Nt
transmit antennas and Nrreceive antennas as illustrated in
Fig. 5. The input n-bit sequence b= [b0b1·· · bn−1]∈Bnis
mapped to a symbol vector s= [s0s1··· sNt−1]∈CNt×1,
where stfor 0≤t≤Nt−1denotes a Gray-coded
data symbol specified in 5G NR [36]. We represent this
bit-to-symbol mapper as s=M(b) = M(b0,·· · , bn−1),
which will be defined in detail in Section 4.2. The baseband
received symbols r∈CNr×1is given by
r=1
√Nt
Hcs+σv,(9)
where Hc∈CNr×Ntdenotes the channel matrix, and
v∈CNr×1denotes the additive white Gaussian noise. Here,
Transmitter
Receiver
Fig. 5. System model for MIMO with Nttransmit and Nrreceiver
antennas.
we assume the narrowband Rayleigh flat fading. That is,
each element of Hc,hut, and each element of v,vu, follow
the standard complex Gaussian distribution CN(0,1) for
0≤u≤Nr−1and 0≤t≤Nt−1. The SNR is defined
as γ= 1/σ2because the symbol vector has the power
constraint Eks/√Ntk2
F= E hPNt−1
t=0 |st|2/Nti= 1. The
constellation size, or modulation order, is denoted by Lc,
and the transmission rate is calculated by
n=Ntlog2(Lc)[bit/symbol].(10)
Corresponding to (9), the ideal MLD is performed as
ˆ
b0,·· · ,ˆ
bn−1= arg min
b0,··· ,bn−1
E(b0,·· · , bn−1),(11)
where we have the objective function
E(b0,·· · , bn−1) =
r−1
√Nt
HcM(b0,·· · , bn−1)
2
F
.(12)
From (11), the exhaustive search by a classical computer
requires the computational time complexity of O(2n), which
is equivalent to the query complexity in CD. Both complex-
ities increase exponentially with the transmission rate n.
To mitigate the exponential complexity, a number of low-
complexity detectors have been proposed in the literature.
The classic ZF detector uses the pseudo-inverse matrix of
WZF = (HH
cHc)−1HH
c(13)
and enables independent detection of data symbols as
ˆ
b0,·· · ,ˆ
bn−1=M−1(WZFr),(14)
where M−1(·)denotes the hard-decision symbol-to-bit
demapper. Similarly, the MMSE detector uses
WMMSE = (HH
cHc+σ2I)−1HH
c(15)
and obtains
ˆ
b0,·· · ,ˆ
bn−1=M−1(WMMSEr).(16)
An MMSE-based interference cancelation method has been
adopted in typical wireless standards such as 5G NR. The
performance of ZF or MMSE detector is worse than that of
MLD. In general, low-complexity detectors improve com-
plexity at the sacrifice of performance.
The above system model and detectors are typical and
common in the field of wireless communications. Since we
consider a general MIMO system, the simulation results
June 1, 2022 7
Re
Im
BPSK
0
1Re
Im
QPSK
00
01
10
11
Re
Im
16-QAM
0000 0010
0001 0011
0101 0111
0100 0110
1010 1000
1011 1001
1111 1101
1110 1100
Fig. 6. Constellation for Gray-coded data symbols specified in the 5G
NR standard [36]
given in this paper are the same as those considering a
multicarrier scenario without inter-subcarrier interference,
or an uplink multi-user scenario in which Ntsingle-antenna
user terminals transmit their symbols and these symbols are
received simultaneously at a base station equipped with Nr
antennas.
4.2 Proposed Method to Transform MLD into HUBO
As described in Section 3, the proposed GAS is capable
of solving a real-valued HUBO problem. We transform
the objective function of MIMO MLD (12) into a HUBO
problem. Specifically, we use the relationship between trans-
mission bits and data symbols, which is specified in the 5G
NR standard [36]. The input n-bit sequence is denoted by
b= [b0b1·· · bn−1]∈Bnand the symbol vector is denoted
by s= [s0s1·· · sNt−1]∈CNt. Then, BPSK symbols
s=M2(b)are generated by [36]
st=1
√2[(1 −2bt) + j(1 −2bt)] (17)
and QPSK symbols s=M4(b)are generated by [36]
st=1
√2[(1 −2b2t) + j(1 −2b2t+1)].(18)
Furthermore, 16-QAM symbols s=M16(b)are generated
by [36]
st=1
√10(1 −2b4t+0)[2 −(1 −2b4t+2 )]
+j
√10(1 −2b4t+1)[2 −(1 −2b4t+3 )] (19)
and 64-QAM symbols s=M64(b)are generated by [36]
st=1
√42(1 −2b6t+0)[4 −(1 −2b6t+2 )[2 −(1 −2b6t+4)]]
+j
√42(1 −2b6t+1)[4 −(1 −2b6t+3 )[2 −(1 −2b6t+5)]].
(20)
A similar relationship for 256-QAM is defined in [36] and
its extension for higher modulation orders can be defined
easily. Overall, Fig. 6 shows the Gray-coded data symbols
defined in (17), (18), and (19).
Our proposed objective function is obtained by sub-
stituting M(·)in (12) with (17), (18), (19), or (20), which
contains nnumber of binary variables b0,··· , bn−1. In the
cases of BPSK and QPSK, our objective function results
in a quadratic form since both symbols are represented
by a linear relationship and the MLD (11) contains the
square of the Frobenius norm. In the case of 16-QAM, the
objective function results in a quartic form since the symbols
are represented by a quadratic relationship. Similarly, the
objective function results in a sextic form in the 64-QAM
case.
The use of data symbols specified in 5G NR is not
straightforward since the objective function inevitably con-
tains higher-order terms if the modulation order is 16 or
higher. Thus, in this form, the conventional quantum an-
nealing and quantum approximate optimization algorithm
requires a transformation from HUBO to QUBO, and this
transformation involves an increase in binary variables,
making the problem more difficult. Our proposed approach
is only possible with the aid of the real-valued support of
GAS. Because of GAS, the query complexity is expected to
be reduced from O(2n)to O(√2n).
Example (QPSK): As a specific example, we consider
the QPSK case (18) with Nt=Nr= 2. The objective
function of (12) can be transformed into
E(b0, b1, b2, b3)
=2
1
X
u=0
1
X
t=0
(Re(hutr∗
u)b2t−Im(hutr∗
u)b2t+1)
+2a1(b0b2+b1b3)+2a2(b0b3−b1b2)
−(a1+a2)(b0+b3)−(a1−a2)(b1+b2),(21)
where we have a1= Re(h00h∗
01) + Re(h10 h∗
11)and a2=
Im(h00h∗
01)+Im(h10 h∗
11). This function (21) is in a quadratic
form.
Example (16-QAM): Additionally, Fig. 7 exemplifies
a specific quantum circuit for the 16-QAM case with Nt=
Nr= 2, where we have n= 8 qubits for binary variables,
m= 6 qubits for real-valued encoding, and random channel
coefficients. As shown in Fig. 7, the objective function results
in a quartic form.
4.3 Proposed Threshold for Further Speedup
GAS obtains a global minimum solution by updating the
threshold value yiand amplifying the probability ampli-
tudes corresponding to values smaller than the threshold.
The query complexity can be reduced by setting the ini-
tial threshold in a different manner rather than the classic
random sampling, although the asymptotic performance
may not change. In this section, we derive the probability
distribution of the objective function value, and use it to
determine a strict threshold, which enables further speedup.
If the information bits in (12) are estimated correctly, the
minimum value of (12) is the Frobenius norm of additive
noise v∈CNr×1as follows:
Emin =σ2
|{z}
known
Nr−1
X
u=0 |vu|2
|{z }
unknown
.(22)
That is, Emin depends on the noise variance σ2, which is
typically known at the receiver, and instantaneous noise vu,
which is unknown in any case. Since the noise is assumed to
follow the complex Gaussian distribution, the magnitude of
June 1, 2022 8
IQFT
Fig. 7. Quantum circuit corresponding to objective function of 16-QAM detection.
Fig. 8. Cumulative distribution of the minimum of objective function
values (25).
the norm follows the Rayleigh distribution, and its square
follows the exponential distribution. As a result, Emin in (22)
follows the Erlang distribution, whose probability density
function is
f(y) = γNryNr−1e−γy
(Nr−1)! ,(23)
where we have SNR γ= 1/σ2. The corresponding cumula-
tive distribution function (CDF) is given by
F(y) = Pr[Y≤y] = 1 −e−γ y
Nr−1
X
u=0
(γy)u
u!.(24)
As an example, if we consider the case with Nr= 2, the
CDF is calculated as
F(y) = Pr[Y≤y]=1−e−γy(1 + γy)(25)
from (24). Fig. 8 exemplifies CDF (25) when SNR is varied
from γ= 5 to 20 dB. Additionally, the CDF of the simulated
objective function values with Nt= 2 and QPSK is also
plotted. As shown in Fig. 8, if the SNR is sufficient, such
as above 10 dB, the theoretical and simulated values are
identical. Thus, it is possible to know in advance that the
minimum value to be calculated is below a certain thresh-
old, which can be determined with a very high degree of
certainty. Given an SNR γ, theoretical values of (24) can be
used to determine a strict threshold.
From (24), the probability that the threshold yis below
the minimum value is
Pr[Y > y] = e−γy (1 + γy).(26)
Let ˜ybe the threshold to be determined and Pbe a small
constant probability, such as P= 10−3and 10−4. Replacing
yand Pr[Y > y]in (26) with ˜yand Pyields
P=e−γ˜y(1 + γ˜y).(27)
Dividing both sides by −egives
−P
e=−(1 + γ˜y)e−(1+γ˜y).(28)
Then, using the Lambert Wfunction, we obtain
W−1−P
e=−(1 + γ˜y) = −1 + ˜y
σ2,(29)
where W−1(·)denotes the lower branch of the Lambert W
function, i.e., W−1(·)≤ −1and W−1(−1/e) = −1. Finally,
the threshold to be determined is
˜y=σ2
|{z}
known
ν
|{z}
known
,(30)
which is similar to (22), and
ν=−1−W−1−P
e.(31)
Here, νis a positive constant and is calculated once before
running our proposed algorithm. For example, we have ν=
9.23 if P= 10−3and ν= 11.8if P= 10−4.
For further speedup, we opt to use the output of the
MMSE detector (15). In [15], Botsinis et al. proposed the
MMSE-based threshold of
¯y=E(¯
b0),(32)
where we have a rough estimate ¯
b0=M−1(WMMSEr).
Our proposed threshold ˜y, which is simpler than ¯y, can be
used together with ¯y. Specifically, we calculate both ˜yand ¯y
at the beginning of Algorithm 2, set the initial threshold as
the smaller of the two, and initialize the first solution with
¯
b0. Let b0be a random n-bit sequence. The initial threshold
June 1, 2022 9
used for the proposed Algorithm 2 can be summarized as
follows:
y0=
E(b0)(Original GAS [9])
¯y(MMSE-based threshold [15])
˜y(Proposed threshold)
min(¯y, ˜y)(Proposed combination)
.(33)
One problem with the proposed threshold ˜yand the
combination min(¯y, ˜y)is that both may become smaller
than the actual minimum. In this case, since there are no
states of interest, GAS will be in a state where the solution
biin Algorithm 2 is not updated. The probability of this
undesirable event occurring is P, i.e.,
Pr[˜y < Emin] = Pr[min(¯y, ˜y)< Emin] = P, (34)
because we have a relationship Pr[¯y < Emin] = 0. Then,
it can be expected that the proposed threshold ˜ymay
degrade the bit error ratio (BER) significantly if Pis not
appropriate. Specifically, the BER of the proposed threshold
˜yis approximated by
P·0.5 + (1 −P)·BERMLD,(35)
where BERMLD is the BER of MLD. In the proposed
combination method, we initialize the first solution with
the MMSE output ¯
b0. Since the initial threshold min(¯y, ˜y)
becomes smaller than the actual minimum with probability
P, the BER of the proposed combination method is approx-
imated by
P·BERMMSE + (1 −P)·BERMLD,(36)
where BERMMSE is the BER of the MMSE detector. Both
(35) and (36) indicate that the design of Phas no significant
effect as long as it is smaller than BERMLD, which can
be calculated exactly in a closed form in advance. In our
performance analysis, the effect of Pwill be investigated in
Fig. 13.
5 PERFORMANCE AN ALYSIS
In this section, we analyze the number of quantum gates
required by GAS, which is represented as a function of
the numbers of qubits nand m. Then, we investigate the
performance of the proposed formulation in terms of BER
and evaluate the proposed algorithm in terms of the rate of
convergence. Here, both integer approximation and direct
encoding are considered. Finally, we evaluate the effects of
the proposed threshold.
5.1 Algebraic Analysis of the Number of Quantum
Gates
A quantum circuit for GAS is composed of H, X, Z, phase,
controlled-phase gates, and the IQFT. In particular, the
state preparation operator Ayis the most complex part
corresponding to the objective function and is dynamically
configured depending on the threshold y. In the quantum
circuit Ay, the number of controlled-phase gates depends on
the number of terms in the objective function. We therefore
derive the number of terms in the objective function that
correspond to each order in an algebraic manner. Ignoring
the power scaling factor, the objective function of MIMO
MLD (12) is transformed into
Nr−1
X
u=0 |ru−hu0s0−hu1s1− · ·· − hu(Nt−1)sNt−1|2
=
Nr−1
X
u=0
(ru−hu0s0−hu1s1− · ·· − hu(Nt−1)sNt−1)
(ru−hu1s0−hu1s1− · ·· − hu(Nt−1)sNt−1)∗.(37)
Here, we focus on three types of terms: first-order terms
such as −r∗
0h00s0and −r0h∗
00s∗
0, squares of the same symbol
such as |h00|2|s0|2and |h01 |2|s1|2, and products of two
symbols such as h00h∗
10s0s∗
1and h∗
00h10 s∗
0s1.
For example, in the relatively simple QPSK case, squares
of the same symbol result in constant terms because of
(18). First-order terms directly result in first-order terms
with respect to binary variables. Products between two
symbols result in products of binary variables. If Nt= 2 and
Nr= 2, four second-order terms appear: b0b2, b1b3, b0b3,
and b1b2. The number of corresponding terms is equal to
the combination of two choices from Ntantennas, e.g.,
Nt
2!=Nt(Nt−1)
2=n(n−2)
8,(38)
where we have the relationship n=Nt·log2(Lc)=2Nt.
In total, the number of second-order terms is calculated as
4·n(n−2)/8 = n(n−2)/2.
Extending the QPSK case, we counted the number of
terms in the objective function for each modulation order
and derived the number of quantum gates required by GAS.
Table 2 summarizes the derived results, where the quantum
gates were categorized by type. As given in Table 2, the
number of controlled-phase gates mainly depends on the
number of binary variables n. Here, 1-CR represents the
controlled-phase gate, and 2-CR, 3-CR, ··· represent the
multi-controlled-phase gates. Since we have the relationship
n=Nt·log2(Lc), the quantum circuit becomes more com-
plex on the order of the square of the number of antennas
Ntand the modulation order Lc.
We analyze the number of quantum gates in the entire
circuit GLiAy|0in+m, where we have G=AyDAH
yO.
In each iteration, the Grover operator is applied Litimes,
where Liis a uniform random number. Ois composed
of a single Z gate and Dis the Grover diffusion operator,
each of which is repeated Litimes. The other part contains
(2Li+1)(n+m)H gates, (2Li+1)mphase gates, (2Li+1)c
controlled-phase gates, and (2Li+ 1) IQFT, where cis
the number of controlled-phase gates given in Table 2. In
summary, Ntand Lcaffect the number of gates on the
order of square, while mand Liaffect it on the linear order.
Here, the only parameter that can be designed is m. Later,
we investigate whether the real-value support of GAS can
reduce m.
5.2 Effects of Integer Approximation
First, Fig. 9 shows BER of the classic MLD and the pro-
posed formulations that consider the integer approxima-
tion with different accuracies. Specifically, the real values
were multiplied by 1, 3, 10 or 20, and approximated by
June 1, 2022 10
TABLE 2
Number of quantum gates required for Ay(n-bit transmission with m-bit accuracy)
Gate BPSK QPSK 16-QAM 64-QAM
Hn+m=O(n+m)n+m=O(n+m)n+m=O(n+m)n+m=O(n+m)
Rm=O(m)m=O(m)m=O(m)m=O(m)
1-CR nm =O(nm)nm =O(nm)nm =O(nm)nm =O(nm)
2-CR n(n−1)m/2 = O(n2m)n(n−2)m/2 = O(n2m)n(n−3)m/2 = O(n2m)n(n−4)m/2 = O(n2m)
3-CR 0 0 n(n−4)m/2 = O(n2m)n(n−6)m+nm/3 = O(n2m)
4-CR 0 0 n(n−4)m/8 = O(n2m) 5n(n−6)m/6 = O(n2m)
5-CR 0 0 0 n(n−6)m/3 = O(n2m)
6-CR 0 0 0 n(n−6)m/18 = O(n2m)
IQFT 1 1 1 1
1x approx.
3x approx.
20x approx.
10x approx.
Fig. 9. BER comparisons for the QPSK case with Nt=Nr= 2.
rounding them to the nearest integers. As references, the
BER curves of ZF, MMSE, and the real-valued formulation
were also plotted. To analyze the effects of approximation
accuracy, BER values were calculated using the state-of-the-
art optimization solver, IBM CPLEX, instead of quantum
simulations. As shown in Fig. 9, BER performance varied
significantly depending on the approximation accuracy. The
high approximation accuracy leads to large integers, result-
ing in an increase in the number of qubits m. In contrast,
the proposed real-valued formulation achieved the same
performance as the classic MLD. This observation indicates
that the proposed real-valued GAS algorithm has to be
invoked to achieve the quantum speedup of the MIMO
MLD problem.
Next, Fig. 10 shows the average objective function values
when increasing the number of iterations, where iterations
in both CD and QD were considered. We assumed a suffi-
ciently high SNR and the fixed channel matrix given in (39).3
We used the original GAS with a random initial threshold
and terminated the simulation if the objective function value
remained the same more than 20 times in CD. In Fig. 10(a),
real values were multiplied by 3 and rounded down to
integers, and in Fig. 10(b), real values were multiplied by 7
and were approximated. The number of qubits mrequired
for encoding the value E(b)−yiwas set to an integer
3. Note that we observed the same trend for different channel coeffi-
cients and SNRs.
(a) QPSK (3x approximation, n= 4,m= 6 qubits).
(b) 16-QAM (7x approximation, n= 8,m= 8 qubits).
Fig. 10. Average objective function values with the integer approximation
and Nt=Nr= 2.
June 1, 2022 11
Hc=0.748510757437062 −0.014877263039446401j1.3215983896521515 + 0.06298233870206783j
0.6371630706424066 −0.14262155021296025j−0.3888005272494009 −0.15170387681055802j.(39)
sufficient not to overflow, i.e., m= 6 in Fig. 10(a) and
m= 8 in Fig. 10(b). It should be noted again that the
integer approximation requires more qubits to encode the
value. Because quantum simulations with n+m= 16 qubits
were time-consuming, we fixed the input bits to 00110101
in Fig. 10(b), while the bits were generated randomly in
Fig. 10(a). For a clear illustration, we added a constant value
to the objective function so that Emin = 0. It was observed
in Fig. 10(a) that the query complexities of GAS in CD
and QD were almost the same as the exhaustive search of
MLD. By contrast, in Fig. 10(b), GAS exhibited better query
complexities in both CD and QD than did MLD. That is, the
advantage of quadratic speedup improved as the problem
size increased.
5.3 Effects of Direct Encoding
Similar to Fig. 10, Fig. 11 shows the average objective
function values when increasing the number of iterations
in CD and QD, where we used the direct encoding. The
simulation parameters were the same as those used in
Fig. 10 except for the real-valued expression and the number
of required qubits m. Specifically, the number of qubits
m= 6 in Fig. 10(a) was reduced to m= 5 in Fig. 11(a).
Similarly, the number of qubits was reduced from m= 8
to m= 6 in Fig. 11(b). As shown in Fig. 11, the same
trend as in Fig. 10 was observed. The important aspect here
is that almost the same query complexities were achieved
despite the reduction in the number of required qubits m.
Hence, our proposed real-valued GAS is capable of reducing
the size of quantum circuits although it maintains a good
performance.
Depending on the channel coefficients and noise, the
integer approximation requires a different number of qubits.
Since both follow the standard Gaussian distribution, the
probability of 0 is the highest, and to deal with smaller
values, a larger factor must be multiplied to the objective
function, resulting in a larger m. By contrast, the direct
encoding is capable of keeping mconstant. The only dis-
advantage is that the probability amplification of GLmay
become insufficient, which was also demonstrated in Fig. 4.
To investigate the disadvantage of the proposed real-
valued GAS and insufficient amplification, in Fig. 12, we
generated random channel coefficients and investigated the
probability density distribution of the number of queries
required to reach the optimal solution, where the param-
eters were the same as those used in Fig. 10(a) except
for mand mwas minimized depending on the random
channel coefficients. It was observed in Fig. 12 that query
complexities in CD and QD increased compared with the
ideal case. Here, the same trend was observed for different
SNRs. Albeit at this expense, the proposed algorithm could
reach the optimal solution in any case. Note that the integer
approximation with the same mas in the direct encoding
could not be plotted in Fig. 12 because it was unable to
reach the solution in most cases.
(a) QPSK (n= 4,m= 5 qubits).
(b) 16-QAM (n= 8,m= 6 qubits).
Fig. 11. Average objective function values with the direct encoding and
Nt=Nr= 2.
5.4 Effects of Initial Threshold for Further Speedup
Finally, in Fig. 13, we evaluated the proposed initial thresh-
old for GAS described in Section 4.3. Here, we averaged
BER with random channel coefficients and noise, considered
SNR of 20 dB, and assumed idealized quantum circuits to
examine the impact of the initial threshold only. Other pa-
rameters were the same as those used in Fig. 11(a). Fig. 13(a)
shows the number of queries in CD, while Fig. 13(b) shows
these in QD. Note that the vertical axis is BER rather than
the objective function value. Specifically, at the left end of
Fig. 13, BER of 0.5corresponds to the bit errors between
the input bits band the random bits b0, and BER of
6.8×10−3corresponds to the errors between band the
MMSE output ¯
b0. As shown in Fig. 13, in both CD and QD,
the proposed threshold, ˜y, converged to the optimal solution
June 1, 2022 12
Fig. 12. Number of queries required to reach the optimal solution.
(a) CD.
(b) QD.
Fig. 13. BER transition with respect to the number of iterations, where
we used random channel coefficients and SNR = 20 dB.
much faster than the classic random threshold. The slopes in
the random and proposed thresholds differed significantly.
This is because the random threshold ranged from the best
to the worst cases, resulting in slow convergence in some
cases. By contrast, the proposed threshold is determined by
constant factors, Pand SNR, which constantly improved
convergence in many cases.
It was also found in Fig. 13 that the proposed threshold
achieved the best performance for P= 10−4and exhibited
lower performance for P= 10−3. As described in Sec-
tion 4.3, Pequals the probability that GAS is in a state where
the solution is not updated. That is, an event of BER = 0.5
occurred with probability P= 10−3, and it resulted in the
error floor of BER around 10−3. This result indicates that the
parameter Phas no significant impact if it is smaller than
BER. Since the exact BER at a given SNR can be calculated
in a closed form in advance, an appropriate Pcan be also
determined in advance accordingly.
Additionally, in Fig. 13, the proposed threshold com-
bined with the MMSE output achieved a faster convergence
compared with the conventional MMSE only case. This
improvement was greater for CD than for QD. That is, the
proposed threshold is particularly useful for improving the
query complexity in CD. In our simulations, this improve-
ment increased upon increasing SNR, which can be verified
from the results shown in Fig. 8. As confirmed in Fig. 8,
the gap between simulated and theoretical values decreased
upon increasing SNR.
6 CONCLUSIONS AND FUTURE WO RKS
In this paper, we proposed the GAS-based quantum algo-
rithm that supports real-valued HUBO. Then, as an appli-
cation example, we formulated the MIMO MLD as a HUBO
problem. The complexity of MLD exponentially increases
with the transmission rate, and low-complexity detectors
sacrifice the achievable performance. Unlike conventional
studies, we constructed specific quantum circuits instead
of assuming an idealized quantum oracle. This enabled us
to analyze the number of qubits and quantum gates in
an algebraic manner. To further accelerate the algorithm,
we derived the probability distribution of the objective
function value and conceived a unique threshold to sample
better states. Numerical simulations demonstrated that the
proposed algorithm could reduce query complexity in CD
and achieve quadratic speedup in QD.
Since this paper focused on a specific construction
method for quantum circuits and their algebraic analysis,
we considered only the hard-decision MLD, instead of error-
correcting codes and soft-decision decoding for classical
bits, which are common in wireless standards. The error
correction capability improves with increasing code distance
and length. For example, the maximum code length of 5G
NR is 1024 for polar code and 8448 for LDPC. However, with
the current computing resources, it is a challenging task to
represent such a large-scale system as a specific quantum
circuit. The proposed real-valued GAS can be applied to
the soft-decision decoding, which will be addressed in our
future work.
June 1, 2022 13
ACKNOWLEDGEMENT
IBM, CPLEX and Qiskit are trademarks of International
Business Machines Corporation.
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