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# A loop Quantum Approximate Optimization Algorithm with Hamiltonian updating

Fang-Gang Duan  Guangdong Provincial Key Laboratory of Quantum Engineering and
Quantum Materials, School of Physics and Telecommunication Engineering, South
China Normal University, Guangzhou 510006, China  Dan-Bo Zhang
<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of Quantum
Engineering and Quantum Materials, School of Physics and Telecommunication
Engineering, South China Normal University, Guangzhou 510006, China
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research
Institute for Physics, South China Normal University, Guangzhou 510006, China

###### Abstract

Designing noisy-resilience quantum algorithms is indispensable for practical
applications on Noisy Intermediate-Scale Quantum (NISQ) devices. Here we
propose a quantum approximate optimization algorithm (QAOA) with a very
shallow circuit, called loop-QAOA, to avoid issues of noises at intermediate
depths, while still can be able to exploit the power of quantum computing. The
key point is to use outputs of shallow-circuit QAOA as a bias to update the
problem Hamiltonian that encodes the solution as the ground state. By
iterating a loop between updating the problem Hamiltonian and optimizing the
parameterized quantum circuit, the loop-QAOA can gradually transform the
problem Hamiltonian to one easy for solving. We demonstrate the loop-QAOA on
Max-Cut problems both with and without noises. Compared with the conventional
QAOA whose performance will decrease due to noises, the performance of the
loop-QAOA can still get better with an increase in the number of loops. The
insight of exploiting outputs from shallow circuits as bias may be applied for
other quantum algorithms.

##  I Introduction

Recent advances on quantum computing technology advocate a growing interest in
finding quantum algorithms that can be implemented effectively on NISQ devices
[1]. One such strong candidate is variational quantum algorithm (VQA) [2, 3,
4, 5, 6], which leverages up the expressive power of parameterized quantum
circuits and a hybrid quantum-classical optimization procedure for finding
solutions. Variational quantum algorithms have wide applications for solving
quantum many-body systems [7, 8], quantum chemistry [9, 10, 11, 12, 13],
optimization problems [14, 15, 16, 17], and so on. Among them, the quantum
approximate optimization algorithm (QAOA) stands out as a VQA for tackling
hard combinatorial optimization problems [18, 19, 20, 21, 22, 23, 24, 25, 26].

The QAOA applies an alternating evolution of mixer and problem Hamiltonians
with optimized evolution periods that its outputs contain the target solution
with a large weighting. By exploiting quantum effects such as interference,
QAOA is believed to show quantum advantages on combinatorial optimization
problems at intermediate depths [27, 15]. One center challenge for
implementing QAOA, however, is that noises on near-term quantum processors
will be detrimental to the performance. For instance, the recent research of
the Google team shows that QAOA performs worse at depths larger than $p=3$
[16]. Moreover, the non-convex and high-dimensional parameter landscape caused
by large $p$ may bring huge difficulty for optimizing [28, 29]. Without
efficient approaches for optimization, the hybrid algorithms will lose it’s
potential advatages [30].

Those challenges addressed at intermediate depths of QAOA have being tackled
recently, e.g., the issue of noise can be reduced by error mitigation
techniques [31, 32], and some constructive optimization procedures have been
proposed [15, 33]. An alternative approach, however, is to ask whether we can
exploit very shallow QAOA to achieve good performance under noise. As the
circuit depth is very shallow, there is no difficulty of variational parameter
optimization, and effects of noises can also be small. Remarkably, even the
lowest depth ($p=1$) of QAOA has been proved that classical computers can’t
simulate efficiently [27]. This suggests a direction for designing noise-
resilience QAOA using shallow circuits while the quantum computing power may
still be exploited.

Our approach is to explore both the parameterized quantum circuit and the
problem Hamiltonian in QAOA for finding the target solution. In the
conventional QAOA, the Hamiltonian which encodes the problem is usually fixed.
However, it is possible that many different Hamiltonians can have the same
solutions or approximated solutions but may be solved with QAOA at different
degrees of difficulty. Motivated by this, we may first use a shallow QAOA to
get some rough estimations of the solution, and then update the problem
Hamiltonian with biased terms. By iterating the procedure of updating the
problem Hamiltonian and finding the optimized parameters of the quantum
circuit, there is a feedback loop between those two subroutines and the
solution may get better with an increase of loops. In this regard, we may call
it as _loop-QAOA_.

In this work, we propose a kind of noise-resilience quantum approximate
optimization algorithm, the loop-QAOA, for solving combinatorial optimization
problems with very shallow circuits, with a feedback loop between Hamiltonian
updating and parameter optimization. Different strategies for constructing
bias field terms are utilized in the Hamiltonian updating. We test the loop-
QAOA and compare it to conventional QAOA on typical Max-Cut graph instances up
to 12-vertex instances, simulated both with and without noise. The results
suggest that loop-QAOA can be a practical quantum algorithm on noisy quantum
devices, where the conventional QAOA may fail to improve its performance by
increasing the circuit depth. Our work gives a new sight for designing hybrid
quantum-classical algorithms using feedback from shallow quantum circuits.

The paper is organized as follows: we firstly review QAOA and the Max-Cut
problem in Sec. II. Then we will propose loop-QAOA and some strategies for
constructing bias field terms in Sec. III. The simulated results are shown in
Sec. IV. Finally, conclusions are given in Sec. V.

##  II Quantum Approximate Optimization Algorithm revisited

The Quantum Approximate Optimization Algorithm (QAOA) by Farhi et al. is a
hybrid quantum-classical algorithm, designed to produce approximated solutions
for combinatorial optimization problems. The solutions are encoded as the
ground state of $H_{P}$, the problem Hamiltonian or the phase Hamiltonian. The
$H_{P}$ is diagonal in the computational basis. As various hard combinatorial
optimization problems can be encoded in such diagonal Hamiltonians [34], the
QAOA is often considered as a potential method of solving combinatorial
optimization problems on quantum computers. These problems are usually defined
on $N$-bit binary strings $\bm{z}$ = $z_{\text{1}}z_{\text{2}}\cdots z_{N}$.
The goal is to find a string which maximizes the classical objective functions
$C(\bm{z}):\\{+1,-1\\}^{\otimes N}\mapsto\mathds{R}$.

As the first step of the QAOA, the classical objective function is encoded in
a quantum Hamiltonian $H_{P}$ defined on $N$-qubits by replacing each variable
$z_{i}$ with a single-qubit $\sigma_{i}^{z}$. The ground state of $H_{P}$ is
parametrized with a quantum circuit which is constructed as follows. For a
$p$-layer QAOA, the state is initialized as $|+\rangle^{\otimes N}$, where
$|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$. Let
$(\vec{\bm{\beta}},\vec{\bm{\gamma}})=(\beta_{1},\cdots\beta_{p},\gamma_{1}\cdots,\gamma_{p})$
be real postive numbers for the layer $p$. The phase Hamiltonian $H_{P}$ and a
mixer Hamiltonian, often chosen as $H_{M}=\sum_{i}\sigma_{i}^{x}$, are then
applied alternately to generate a variational wavefunction

$|\Psi_{p}(\vec{\bm{\beta}},\vec{\bm{\gamma}})\rangle=\prod_{j=1}^{p}e^{-i\beta_{j}H_{M}}e^{-i\gamma_{j}H_{P}}|+\rangle^{\otimes
N}.$ (1)

The expectation value $H_{P}$ with regard to the variational state is

$F_{p}(\vec{\bm{\gamma}},\vec{\bm{\beta}})=\langle\Psi_{p}(\vec{\bm{\beta}},\vec{\bm{\gamma}})|H_{P}|\Psi_{p}(\vec{\bm{\beta}},\vec{\bm{\gamma}})\rangle,$
(2)

which can be obtained by measurements in the computational basis. The goal is
to make $F_{p}(\vec{\bm{\gamma}},\vec{\bm{\beta}})$ minimal. The parameters
$\beta_{1},\cdots\beta_{p},\gamma_{1}\cdots,\gamma_{p}$ are adjusted in
repeated learning loops in a classical computer to minimize the objective
function Eq. (2) evaluated with a quantum computer.

We use the Max-Cut to illustrate the QAOA more concisely. Consider a graph $G$
with $N$ vertices over a vertex set $V$ and edge set $E$. The Max-Cut problem
is to divide the vertexes $V$ into two disjoint subsets such that the number
of cuts (edges connecting two disjoint subsets) is maximized. Let
$\omega_{ij}$ be the weight of the associated edge $\left<i,j\right>\in E$.
The objective function can be defined as a sum of edge terms in the graph as

$C(\bm{z})=\sum_{\left<i,j\right>\in
E}C_{ij}=\frac{1}{2}\sum_{\left<i,j\right>\in E}\omega_{ij}(1-z_{i}z_{j})$ (3)

where $z_{i}=\pm 1$. By maximizing $C(\bm{z}$, the cut set $S(z)\in E$ can be
obtained for the Max-Cut problem.

The Max-Cut Problem is in the APX-complete complexity class, meaning that
there is no polynomial-time approximation scheme to find the optimal solution,
unless P = NP [35, 36, 37]. But we may find a polynomial-time approximation
algorithm with a solution of a string $\bm{z^{\prime}}$ that achieves a
desired approximation ratio

$\frac{C(\bm{z^{\prime}})}{\max\\{C(\bm{z})\\}}\geq r^{*}.$ (4)

The objective function of the Max-Cut problem can be encoded as a problem
Hamiltonian,

$H_{P}=-\frac{1}{2}\sum_{\left<i,j\right>\in
E}\omega_{ij}(1-\sigma_{i}^{z}\sigma_{j}^{z}).$ (5)

Note an addition minus sign in $H_{P}$ turns the task of maximizing the
objective function $C(\bm{z})$ to finding the lowest energy of $H_{P}$. If
spins $\sigma_{i}^{z}$ and $\sigma_{j}^{z}$ are anti-aligned, an edge
$\left<i,j\right>$ contributes with an energy $-\omega_{ij}$; otherwise, the
energy contribution is zero.

It has been proven NP-hard to design an algorithm that guarantees an
approximation ratio of $r^{*}\geq 16/17\approx 0.941$ for Max-Cut on all
graphs [35]. The best-known classical approximation algorithm, of Goemans and
Williamson (GW) [38], guarantees an approximation ratio of $r^{*}\approx
0.878$ using semidefinite programming and randomized rounding. For more
special graphs the GW can do better, $r^{*}$ can reach 0.932 for u3R graphs
[39]. Farhi et al. [14] showed that QAOA at level $p=1$ the worst-case
approximation ratio is 0.6924 for u3R graphs. It is known the performance of
QAOA gets better with an increase of the depth $p$ [14]. The goal of quantum
approximate optimization is to raise $r^{*}$ that is beyond the capacity of
classical algorithms.

The QAOA often requires quantum circuits of intermediate depths to achieve the
desired accuracy for combinatorial optimization problems. However, the benefit
of increasing $p$ in QAOA will be hampered due to noises in quantum
processors. The performance in the presence of noises may get worse for $p$
larger than a certain small value [16]. Even without noise, optimization of
variational parameters for large $p$ can be a great challenge. The two issues
can be avoided if a shallow circuit is used in QAOA, which would be explored
in the next section.

##  III Loop-QAOA with problem Hamiltonian updating

In QAOA, it is both the problem Hamiltonian and the parameterized quantum
circuit to determine the solution, as seen in Eq. (2). Usually, the problem
Hamiltonian is given beforehand, and what is left is to increase the depth $p$
or design novel variational ansatz. Nevertheless, there is plenty of room for
handling the problem Hamiltonian. Since the same solution may correspond to
different problem Hamiltonians. It is possible that the problem Hamiltonian
can be transformed to a new one with the same ground state but easier to
solve. Instead of increasing the depth $p$, one can fix QAOA with a very small
$p$, while the problem Hamiltonian is improved. Guided by this insight, we
propose the loop-QAOA, where the problem Hamiltonian is updated in an
iterative fashion with the feedback from shallow-circuit QAOA.

Let us first elaborate on the case of one loop to illustrate the mechanism
behind loop-QAOA. Throughout the paper, we use the least shallow circuit in
loop-QAOA, namely $p=1$, while the case $p>1$ can be investigated similarly.
In practice, the $p=1$ QAOA can show much better performance than randomly
guessing. Moreover, it gives a probability distribution over all possible
bitstrings. The output of $p=1$ QAOA thus can reveal the underlying structure
of the problem to some degree, which can be exploited as a bias for the
combinatorial optimization problems. For instance, the bitstrings with high
probabilities are informative for constructing bias terms for updating the
problem Hamiltonian.

The procedure of training $p=1$ QAOA and updating the problem Hamiltonian can
be iterated multiple times. Schematic of the loop-QAOA can be found in Fig.
1a. The aim is to gradually transform the Hamiltonian that the chance to get
the target solution will increase with more loops. To make this possible, the
bias terms and the scheme of Hamiltonian updating should be designed properly.
We note that using bias terms to accelerate the quantum optimization has been
considered in the context of quantum annealing [40].

(a)

(b)

Figure 1: Schematic of a $p$-level QAOA and loop-QAOA strategy. (a) The
circuit diagram in the solid frame represents QAOA. A quantum circuit takes
input $|+\rangle^{\otimes N}$ and alternately applies
$U(\beta_{i})=e^{-i\beta_{i}H_{P}}$ and $U(\gamma_{i})=e^{-i\gamma_{i}H_{M}}$,
and measure the final state to obtain expectation value with respect to the
objective function $C(z)$. This result is optimized by a classical optimizer
to get the maximum value. The dashed box represents the loop-QAOA strategy. We
add bias field terms to objective Hamiltonian through the measurement results
of each loop. (b) Schematic of loop-QAOA strategy applied to a 6-vertex w3R
graph, where weights are updated at each loop.

We illustrate the procedure of Hamiltonian updating with bias terms with the
Max-Cut problem. Suppose we have a connected weighted graph $G=(V,E)$ with $N$
nodes, and the weight set of its edges is
$W=\\{\omega_{ij}:\left<i,j\right>\in E\\}$. The output of $p=1$ QAOA is a
probability distribution $Q=\\{(\bm{z}_{i},p_{i})\\}$, where $\bm{z}_{i}$ is
the bitstring and $p_{i}$ is the corresponding probability. We will choose the
results with $p_{i}$ greater than a certain threshold $p^{*}$, which are
informative for constructing bias terms.

The construction of bias terms is based on how the distribution of bitstrings
output from QAOA contributes to the objective function. Consider a string
$\bm{z}_{1}$ for instance. If two of the spins are opposite,
$\bm{z}_{1i}=0,\bm{z}_{1j}=1$ or $\bm{z}_{1i}=1,\bm{z}_{1j}=0$ (based on the
$\mathds{Z}_{2}$ symmetry of the Max-Cut problem), then the $\left<i,j\right>$
edge will make a contribution of $\omega_{ij}$ to the objective function. If
the spins are the same, namely $\bm{z}_{1i}=\bm{z}_{1j}$, the contribution is
$0$. Considering the distribution from measurement results, $\left<i,j\right>$
in $z_{1}$ makes a contribution of $p_{z_{1}}\omega_{ij}$. Then the
contribution of the edge $\left<i,j\right>$ to the objective function in total
is

$C_{ij}=\sum_{k=1}^{N}p_{z_{k}}\omega_{ij}$ (6)

After going through all the edges in $G$, we get a set $\\{C_{ij}\\}$. We can
reward the edge with a larger contribution and punish the edge with a smaller
contribution.

The reward or punishment is realized by adding bias terms to the problem
Hamiltonian. For this, we introduce a mapping $F:W\mapsto W^{\prime}$, which
tunes the weights of the graph. As a schematic, Fig. 1b shows how weights of a
6-nodes w3R graph are updated in the loop-QAOA, which finally becomes easy to
solve. There can be different mapping schemes. Apparently, different mapping
methods or different mapping parameters will lead to different results. Here,
we list some bias field methods: (1). Bias field and antibias field:
$W^{\prime}=\\{\omega_{ij}\pm kC_{ij}\\}$($k$ can be a constant or a function,
for controling the bias field terms.); (2). Fixed bias field:
$W^{\prime}=\\{\omega_{ij}+\text{fixed bias terms}\\}$. We adopt a modified
version of the first scheme with bias fields that updates edge weight in each
loop via the following mapping,

$\omega_{ij}^{\prime}=(1-\tau\sum^{N}_{z_{ki}=z_{kj}}{p_{\bm{z}_{k}}})\omega_{ij},$
(7)

where $\tau$ is a parameter to control the strength of bias.

Using this scheme, the solution space will be reduced by punishing a bitstring
$\bm{z}_{k}$ if there are many pairs of spins that are the same
$z_{ki}=z_{kj}$ on the edges. Note that the weights may approach zeroes for a
large number of loops. To avoid this situation, we can control the bias
strength $\tau$ to make the strategy get the desired results with a small
number of loops. A heuristic formula of bias strength is proposed by
interpolation:

$\tau=\frac{1}{2^{n-2}}[1-\frac{1}{3}(\frac{1}{g\Delta_{max}})^{\frac{1}{l+f(n)}}-\frac{2}{3}(\frac{1}{g\Delta_{min}})^{\frac{1}{l+f(n)}}]$
(8)

where $n$ is nodes number, $l$ is the current number of loops, $f(n)$ is a
function monotonous with the number of nodes, $\Delta_{max}$ is the maximum
weight difference, and $\Delta_{min}>0$ is the minimum weight difference. The
factor $g$ ensures that the results are within the given accuracy range, and
we chose $g=10^{3}$ to keep three significant digits. The power correspond to
current loops of algorithm, where $f(n)$ is usually taken as a fixed constant.
$\Delta_{max}$ and $\Delta_{min}$ ensure that all edges will not become too
small within a certain number of loops. The coefficinet $1/3$ and $2/3$ are
obtained by interpolating, and can be adjusted to other numbers. We mention
that the above method is not only limited to the Max-Cut problem but also
effective for general combinatorial optimization problems.

(a)

(b)

Figure 2: Performances of the conventional QAOA and the loop-QAOA with two
different choices of bias strengths. (a) the average success rates $p_{s}$ on
$10^{4}$ randomly generated 12-vertex w3R graphs. $\tau$ indicates the bias
strength, and they are chosen by the interpolation in Eq. (8), or randomly
choosing form $2.0\times 10^{-4}$ to $2.8\times 10^{-4}$. For all cases loop-
QAOA gradually approaches the conventional QAOA with the increase of the
number of loops. (b) The average approximation ratio $r^{*}$ of the QAOA and
the loop-QAOA with the interpolation method.

##  IV Simulation Results

In this section, we first give the simulation methods and then present the
simulation results. Comparisons between the performances of loop-QAOA and the
conventional QAOA are given, both in the presence or absence of noises.

###  IV.1 Simulation methods

Without losing the difficulty of the combinatorial optimization problem, we
are concerned about connected graphs with weights (such as w3R) $w_{ij}$
chosen uniformly at random from the interval $[0,1]$. We consider randomly
generated instances w3R graphs with vertex number $8\leq N\leq 14$. For each
vertex number, $10^{4}$ w3R graph instances are generated. The loop-QAOA
strategy at each loop and the conventional QAOA at each layer $p$ are applied
for those generated graphs. As for the optimization method, we apply both the
gradient-based optimization routine (BFGS) in our numerical simulation. It is
noted that non-gradient-based routines such as Nelder-Mead [41, 42] and
Bayesian methods [43] also works for the loop-QAOA.

We also add noise in our numerical simulation to address that the loop-QAOA
can be noise-resilience. We choose three quantum noise models to test the
algorithm, namely quantum bit flip channel, quantum phase flip channel, and
depolarizing channel [44, 45]. The quantum bit flip channel evolves a density
matrix via:

$\rho\to(1-q)\rho+qX\rho X,$ (9)

where $q$ is the probability of a bit flip ($q$ instead of $p$ is to
distinguish it from the previous $p$-layer). The quantum phase channel evolves
a density matrix via:

$\rho\to M_{0}\rho M_{0}^{\dagger}+M_{1}\rho M_{1}^{\dagger}$ (10)

with :

$M_{0}=\sqrt{q}\begin{bmatrix}1&0\\\
0&1\end{bmatrix}\quad,M_{1}=\sqrt{1-q}\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}\quad,$ (11)

where $q$ is the probability of a phase flip. The depolarizing channel applies
one of $4^{n}$ disjoint possibilities: the identity channel or one of the
$4^{n}-1$ pauli gates. The probability of a non-identity Pauli gates are
$q/(4^{n}-1)$, and the identity is done with probability $1-q$, where $q$ is
the probability that one of the Pauli gates is applied, $n$ is the number of
qubits, $P_{i}$ are the $4^{n}-1$ Pauli gates(excluding the identity). This
channel evolves a density matrix via

$\rho\to(1-q)\rho+\frac{q}{4^{n}-1}\sum_{i}P_{i}\rho P_{i}$ (12)

We apply these models to the whole quantum circuit. We show results for the
case of $q=0.1$ in bit flip channel and phase flip channel, $q=0.05$.in
depolarizing channel in Fig 3.

(a) Bit flip channel

(b) Phase flip channel

(c) Depolarizing channel

Figure 3: Performances of the conventional QAOA and the loop-QAOA at three
different noise model environment. (a),(b) and (c) shows the average success
rates $p_{s}$ on $10^{4}$ randomly generated 12-vertex w3R graphs with bit
flip channel, phase flip channel and depolarizing channel, respectively.

###  IV.2 Results

We first compare the performances of the conventional QAOA and the loop-QAOA
in the absence of noises. For the loop-QAOA, the bias strengths at different
loops are chosen according to the interpolation in Eq. (8) or chosen randomly
from a given interval. The results on a set of 12-vertex w3R graphs with
$10^{4}$ instances are shown in Fig. 2. Since QAOA can be regarded as the case
where the bias constant $\tau=0$, the performance of the two is almost the
same here at $p=1$ and loop$\ =1$. At low $p$ or low loops, the loop-QAOA
strategy is slightly inferior to the original QAOA method. However, after a
certain number of loops, the performance of loop-QAOA can gradually approach
the original QAOA. When loop $\geq 8$, the performance of the two is very
close. For the conventional QAOA, this method yields a success rate of
$p_{s}=0.983$ at $p=10$. The interpolation loop-QAOA reaches a success rate of
$p_{s}=0.979$, which is very close to the former. On the other hand, the loop-
QAOA with random choices of bias strengths performs worse than the
interpolation one. The above results indicate that the loop-QAOA can be a
potential powerful quantum approximate optimizer using only a very shallow
quantum circuit as long as appropriate bias field parameters are selected.

Remarkably, the loop-QAOA can have an advantage over the conventional QAOA
when noises should be considered. Under three noise models as described in Eq.
(9) (10) (12), we simulate the QAOA and the loop-QAOA on randomly generated
$10^{3}$ instances of w3R graphs with vertex number $N=4$, and the results are
shown in Fig. 3. We limit $p$ to be less than or equal to 5 for consideration
of simulation time and cost. Due to the existence of noise, the performance of
QAOA improves slower with the increase of $p$ and even can decrease for larger
$p$. However, the performance of the loop-QAOA still raises with the loops and
finally reaches a very high success rate, which significantly outperforms the
best performance of the conventional QAOA.

##  V Conclusion

To conclude, we have proposed the loop-QAOA as a noise-resilience quantum
algorithm for solving combinatorial optimization problems. The loop-QAOA
exploits the outputs from a very shallow circuit as a bias to update the
problem Hamiltonian and forms a feedback loop between Hamiltonian updating and
parameter optimization. We have tested the loop-QAOA and compared it to the
conventional QAOA on the Max-Cut problem with and without noises. Remarkably,
the performance of the loop-QAOA under noises can still be made better by
increasing the loop number, while the performance of the conventional QAOA
will cease to increase at a small depth. Our work points out a direction for
designing quantum algorithms with practical applications on NISQ devices with
very shallow circuits.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grant No.12005065), by the Guangdong Basic and Applied Basic Research Fund
(Grant No.2021A1515010317), and the Key Project of Science and Technology of
Guangzhou(Grant No.2019050001).

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