量子计算中的短深度算法：超越 Swap Test
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"Learning the quantum algorithm for state overlap.pdf文件探讨了如何使用机器学习方法来开发量子计算机上的短深度算法，特别是在计算两个量子状态之间重叠的场景下，即Tr(ρσ)。标准的计算方法是Swap Test，常用于量子支持向量机等应用，并能度量Rényi纠缠。该研究发现了一些比Swap Test具有更短深度的算法，其中一种甚至具有与问题规模无关的常数深度。此外，研究还针对Rigetti和IBM量子计算机的特定硬件连接性和门集进行了优化，展示了缩短算法能在实际量子硬件上有效执行的优势。"
本文着重讨论了在近似量子计算机中，由于退相干和门不精确性等问题，设计短深度算法的重要性。短深度算法能有效减少计算错误，提高系统性能。作者提出了一种基于机器学习的方法来探索这种算法，这种方法对于发现新的、效率更高的量子算法具有开创性意义。
在量子计算中，计算两个量子状态ρ和σ之间的重叠是常见的任务，这在量子信息处理和量子机器学习中有多种应用。传统的Swap Test算法虽然有效，但在某些情况下可能深度较大，导致误差增加。通过机器学习技术，研究者找到了替代方案，这些新算法在保持功能的同时，显著降低了所需的量子门深度，这有助于减少因量子比特的退相干而引入的错误。
特别地，研究者针对特定的量子计算机硬件，如Rigetti和IBM的设备，优化了这些短深度算法。考虑到实际量子处理器的物理连接性和可用的门集，这些优化使得提出的算法更加实用，能在真实世界的应用中提供更好的性能。这表明，结合机器学习和硬件理解可以推动量子计算领域的进步，为解决实际问题提供更高效、适应性强的量子算法。
总结来说，"Learning the quantum algorithm for state overlap.pdf"文件揭示了机器学习在优化量子计算算法中的潜力，特别是在减少量子状态重叠计算的深度方面。这项工作不仅提供了理论上的新见解，还展示了如何将这些理论应用于现实世界的量子计算机，从而提升其性能和可靠性。
3
FIG. 2: Swap Test circuits. (A) The canonical Swap Test
circuit. H indicates the Hadamard gate. (B) The Swap Test
circuit adapted for IBM’s 5-qubit quantum computer, con-
structed by decomposing controlled-swap into the Toffoli gate,
via Refs. [34, 35], and then manually eliminating gates that
had no effect on the output. T is the π/8 phase gate. (C)
The structure of a Swap Test circuit, showing the locations of
the one-qubit gates and controlled-Z gates, constructed au-
tomatically by Rigetti’s compiler for their 19-qubit quantum
computer. Appendix A gives the full specification of that
circuit.
algorithms on Rigetti’s and IBM’s quantum computers,
leading to a reduction in the computational error relative
to the Swap Test.
II. MACHINE-LEARNING APPROACH
Our machine-learning approach is summarized in
Fig. 1. The variables are divided up into the hyper-
parameters (i.e. the “resources”) and the optimization
parameters (i.e. the “algorithm”).
A. Resources
The hyperparameters are the quantum resources of the
circuit. At the input, the resources are the number of
ancilla qubits and data qubits that store the input data
for the computation. At the output, the resources are
the locations of the measurements (see Fig. 1). As an
example, in the Swap Test for single-qubit states, we are
allowed access to one ancilla qubit and two data qubits
at the input, and we can measure only the ancilla qubit
at the output.
The input data may be classical or quantum, depend-
ing on the computation of interest. In the case of state
overlap, the input data are quantum states and hence
no encoding is necessary. However, for completeness, we
note that our approach also applies to classical inputs, in
which case the encoding (i.e., storing the classical data
in the quantum state of the data qubits) can be treated
as a hyperparameter that one fixes while optimizing the
algorithm.
B. Algorithm
Our approach searches for an optimal algorithm, where
we consider the algorithm to be a quantum gate sequence
with associated classical post-processing. We parameter-
ize (and hence optimize over) both the gate sequence and
the post-processing.
Let us first consider the gate sequence. We define a
gate set A = {A
j
(θ)}. Here, each gate A
j
is either a
one-qubit or two-qubit gate and may also have an inter-
nal continuous parameter θ. Hence, A is a discrete set,
but each element of A may have a continuous parame-
ter associated with it. The precise choice of A depends
on which hardware one is considering. For example, the
connectivity differs between IBM and Rigetti hardware,
and the former employs CNOT gates while the latter
employs controlled-Z gates. For IBM’s 5-qubit computer
“ibmqx4” we can write out the gate set as
A
ibmqx4
= {CNOT
10
, CNOT
20
, CNOT
21
, CNOT
32
,
CNOT
24
, CNOT
34
, U
0
(θ), U
1
(θ), U
2
(θ),
U
3
(θ), U
4
(θ)}, (1)
where U
j
(θ) is an arbitrary gate on qubit j and CNOT
jk
is a CNOT from control qubit j to target qubit k. An-
gles θ in Eq. (1) may be encoding multiple parameters.
In this article, we treat all one-qubit gates equally in
the sense that all one-qubit gates are equally complex to
implement, although our approach could easily be gener-
alized to account for different complexities for different
one-qubit gates.
We consider a generic sequence of d gates,
G
~
k
(
~
θ) = A
k
d
(θ
d
) · ·· A
k
2
(θ
2
)A
k
1
(θ
1
) , (2)
where
~
k = (k
1
, ..., k
d
) is the vector of indices describ-
ing which gates are employed in the gate sequence and
~
θ = (θ
1
, ..., θ
d
) is the vector of continuous parameters
associated with these gates.
The measurement results give rise to an outcome prob-
ability vector ~p = (p
1
, ..., p
l
, ...). The desired output
might be one of these probabilities p
l
, or it might be
some simple function of these probabilities. Hence, we
allow for some simple classical post-processing of ~p in or-
der to reveal the desired output. While there is enormous
freedom in applying a function to ~p, we consider a simple
linear combination of probabilities:
g(~p) = ~c · ~p =
X
l
c
l
p
l
(3)
where ~c is a vector of coefficients whose elements are cho-
sen according to c
l
∈ {−1, 0, 1}. This post-processing is
sufficient for the application in this paper (state overlap),
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