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added reference CLZ22 and improved organization
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Expand Up @@ -1171,56 +1171,6 @@ <h2>Approximation and Simulation Algorithms</h2>
polynomial speedup for evaluating partition functions is given in [<a href="#M15b">265</a>].
<br /><br />

<b>Algorithm:</b> Quantum Approximate Optimization <br />
<b>Speedup: </b> Superpolynomial <br />
<b>Implementation:</b> <a href="https://short.classiq.io/qaoa">Classiq</a>, <a href="https://github.com/quantumlib/Cirq/blob/main/examples/qaoa.py">Cirq</a><br />
<b>Description:</b> For many combinatorial optimization problems,
finding the exact optimal solution is NP-complete. There are also hardness-of-approximation
results proving that finding an approximation with sufficiently
small error bound is NP-complete. For certain problems there is a gap
between the best error bound achieved by a polynomial-time classical
approximation algorithm and the error bound proven to be NP-hard. In
this regime there is potential for exponential speedup by quantum
computation. In [<a href="#FGG14a">242</a>] a new quantum algorithmic
technique called the Quantum Approximate Optimization Algorithm (QAOA)
was proposed for finding approximate solutions to
combinatorial optimization problems. In [<a href="#FGG14b">243</a>] it
was subsequently shown that QAOA solves a combinatorial
optimization problem called Max E3LIN2 with a better approximation
ratio than any polynomial-time classical algorithm known at the
time. However, an efficient classical algorithm achieving an even
better approximation ratio (in fact, the approximation ratio
saturating the limit set by hardness-of-approximation) was
subsequently discovered [<a href="#BMO15">260</a>]. Presently, the power of
QAOA relative to classical computing is an active area of research
[<a href="#LZ16">300</a>,<a href="#WHT16">301</a>,<a href="#FH16">302</a>,
<a href="#Chamon">314</a>,<a href="#SLC24">451</a>,<a href="#BFM22">452</a>,<a href="#BM22">476</a>].
<br /><br />

<a name="semidefinite"></a>
<b>Algorithm:</b> Semidefinite Programming <br />
<b>Speedup: </b> Polynomial (with some exceptions)<br />
<b>Description:</b> Given a list of <i>m</i> + 1 Hermitian \(n \times n \)
matrices \(C, A_1, A_2, \ldots, A_m\) and <i>m</i> numbers
\(b_1, \ldots, b_m \), the problem of semidefinite programming is to
find the positive semidefinite \( n \times n \)
matrix <i>X</i> that maximizes tr(<i>CX</i>) subject to the
constraints \( \mathrm{tr} (A_j X) \leq b_j \) for \( j = 1,2,\ldots,
m \). Semidefinite programming has many applications in operations
research, combinatorial optimization, and quantum information, and it
includes linear programming as a special case. Introduced in
[<a href="#BS16">313</a>], and subsequently improved in [<a href="#BKL17">383</a>, <a href="#AGG20">425</a>],
quantum algorithms are now known that can approximately solve semidefinite programs to within \( \pm \epsilon \) in time
\( O (\sqrt{m} \log m \cdot \mathrm{poly}(\log n, r, \epsilon^{-1})) \), where <i>r</i> is
the rank of the semidefinite program. This constitutes a quadratic speedup over the fastest
classical algorithms when <i>r</i> is small compared to <i>n</i>. The quantum algorithm is
based on amplitude amplification and quantum Gibbs sampling [<a href="#Poulin_Wocjan">121</a>, <a href="#CS16">307</a>].
In a model in which input is provided in the form of quantum states the quantum algorithm
for semidefinite programming can achieve superpolynomial speedup, as discussed in [<a href="#BKL17">383</a>],
although recent dequantization results [<a href="#CGL20">421</a>] delineate limitations on
the context in which superpolynomial quantum speedup for semidefinite programs is possible.
<br /><br />

<b>Algorithm:</b> Zeta Functions <br />
<b>Speedup:</b> Superpolynomial <br />
<b>Description:</b> Let <i>f</i>(<i>x</i>,<i>y</i>) be a degree-<i>d</i>
Expand Down Expand Up @@ -1411,6 +1361,32 @@ <h2>Optimization, Numerics, and Machine Learning</h2>
preparation.
<br /><br />

<b>Algorithm:</b> Quantum Approximate Optimization <br />
<b>Speedup: </b> Superpolynomial <br />
<b>Implementation:</b> <a href="https://short.classiq.io/qaoa">Classiq</a>, <a href="https://github.com/quantumlib/Cirq/blob/main/examples/qaoa.py">Cirq</a><br />
<b>Description:</b> For many combinatorial optimization problems,
finding the exact optimal solution is NP-complete. There are also hardness-of-approximation
results proving that finding an approximation with sufficiently
small error bound is NP-complete. For certain problems there is a gap
between the best error bound achieved by a polynomial-time classical
approximation algorithm and the error bound proven to be NP-hard. In
this regime there is potential for exponential speedup by quantum
computation. In [<a href="#FGG14a">242</a>] a new quantum algorithmic
technique called the Quantum Approximate Optimization Algorithm (QAOA)
was proposed for finding approximate solutions to
combinatorial optimization problems. In [<a href="#FGG14b">243</a>] it
was subsequently shown that QAOA solves a combinatorial
optimization problem called Max E3LIN2 with a better approximation
ratio than any polynomial-time classical algorithm known at the
time. However, an efficient classical algorithm achieving an even
better approximation ratio (in fact, the approximation ratio
saturating the limit set by hardness-of-approximation) was
subsequently discovered [<a href="#BMO15">260</a>]. Presently, the power of
QAOA relative to classical computing is an active area of research
[<a href="#LZ16">300</a>,<a href="#WHT16">301</a>,<a href="#FH16">302</a>,
<a href="#Chamon">314</a>,<a href="#SLC24">451</a>,<a href="#BFM22">452</a>,<a href="#BM22">476</a>].
<br /><br />

<a name="gradients"></a>
<b>Algorithm:</b> Gradient Estimation and Learning Polynomials<br />
<b>Speedup:</b> Polynomial <br />
Expand All @@ -1433,6 +1409,30 @@ <h2>Optimization, Numerics, and Machine Learning</h2>
queries. See also: <a href="#convex_optimization">convex optimization</a>.
<br /><br />

<a name="semidefinite"></a>
<b>Algorithm:</b> Semidefinite Programming <br />
<b>Speedup: </b> Polynomial (with some exceptions)<br />
<b>Description:</b> Given a list of <i>m</i> + 1 Hermitian \(n \times n \)
matrices \(C, A_1, A_2, \ldots, A_m\) and <i>m</i> numbers
\(b_1, \ldots, b_m \), the problem of semidefinite programming is to
find the positive semidefinite \( n \times n \)
matrix <i>X</i> that maximizes tr(<i>CX</i>) subject to the
constraints \( \mathrm{tr} (A_j X) \leq b_j \) for \( j = 1,2,\ldots,
m \). Semidefinite programming has many applications in operations
research, combinatorial optimization, and quantum information, and it
includes linear programming as a special case. Introduced in
[<a href="#BS16">313</a>], and subsequently improved in [<a href="#BKL17">383</a>, <a href="#AGG20">425</a>],
quantum algorithms are now known that can approximately solve semidefinite programs to within \( \pm \epsilon \) in time
\( O (\sqrt{m} \log m \cdot \mathrm{poly}(\log n, r, \epsilon^{-1})) \), where <i>r</i> is
the rank of the semidefinite program. This constitutes a quadratic speedup over the fastest
classical algorithms when <i>r</i> is small compared to <i>n</i>. The quantum algorithm is
based on amplitude amplification and quantum Gibbs sampling [<a href="#Poulin_Wocjan">121</a>, <a href="#CS16">307</a>].
In a model in which input is provided in the form of quantum states the quantum algorithm
for semidefinite programming can achieve superpolynomial speedup, as discussed in [<a href="#BKL17">383</a>],
although recent dequantization results [<a href="#CGL20">421</a>] delineate limitations on
the context in which superpolynomial quantum speedup for semidefinite programs is possible.
<br /><br />

<a name="convex_optimization"></a>
<b>Algorithm:</b> Convex Optimization<br />
<b>Speedup:</b> Polynomial <br />
Expand Down Expand Up @@ -1652,6 +1652,17 @@ <h2>Optimization, Numerics, and Machine Learning</h2>
making a connection between Nash equilibria and Gibbs sampling.
<br /><br />

<b>Algorithm:</b> Lattice Problems by Filtering<br />
<b>Speedup:</b> Exponential<br />
<b>Description:</b> We are given a set of basis vectors in \( \mathbb{Z}^n \) and we consider the lattice \( \mathcal{L} \) defined by all integer linear combinations
of these basis vectors. In general, finding the shortest nonzero vector in \( \mathcal{L} \) is an NP-hard problem. Given a target point \( \mathbf{x} \in \mathbb{Z}^n \), finding the
nearest element of \( \mathcal{L} \) is also NP-hard. There are various problems of finding approximately optimal solutions, or finding solutions given certain promises on the $\mathcal{L}$
that are not known to be NP-hard yet lack known polynomial time classical solutions. Such problems, versions of which underlie many post-quantum cryptosystems, provide very important
targets for quantum algorithms research. In [<a href="#CLZ22">498</a>], Chen, Liu, and Zhandry obtained polynomial time quantum algorithms for some lattice problems that do not have known polynomial-time classical solutions. These quantum algorithms apply in a parameter regime that does not break the any of the main candidate post-quantum cryptosystems. A key ingredient
in these algorithms is the observation from [<a href="#Regev_lattice">78</a>, <a href="#Aharonov_Tashma">5</a>]. that the quantum Fourier transform allows efficient reductions in both directinos between shortest vector problems on a lattice and nearest vector problems on the dual lattice. In [<a href="#CLZ22">498</a>], the authors add also a new ingredient,
which is a filtering technique that uses quantum measurements in nontrivial bases to improve the parameters of these reductions in such a way as to achieve quantum advantage.
<br /><br />

<hr />

<a name="acknowledgments"></a>
Expand Down Expand Up @@ -5422,6 +5433,14 @@ <h2>References</h2>
<br /><em><a href="https://arxiv.org/abs/2110.13086">arXiv:2110.13086</a></em>, 2021.
<br /><br /></dd>

<dt><a name="CLZ22">498</a></dt>
<dd>
Yilei Chen, Qipeng Liu, and Mark Zhandry
<br />Quantum algorithms for variants of average-case lattice problems via filtering
<br /><em>Proceedings of EUROCRYPT22</em>, 372 - 401, 2022.
<br />[<a href="https://arxiv.org/abs/2108.11015">arXiv:2108.11015</a>]
<br /><br /></dd>

</dl>
</div>
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