Testing quantum computers with the protocol of quantum state matching Adrian Ortega1Orsolya K alm an1yand Tam as Kiss1z

2025-05-02 0 0 687.37KB 26 页 10玖币
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Testing quantum computers with the
protocol of quantum state matching
Adrian Ortega,1, ∗Orsolya K´alm´an,1, †and Tam´as Kiss1, ‡
1Wigner RCP, Konkoly-Thege M. u. 29-33, H-1121 Budapest, Hungary
(Dated: February 15, 2023)
The presence of noise in quantum computers hinders their effective operation.
Even though quantum error correction can theoretically remedy this problem,
its practical realization is still a challenge. Testing and benchmarking noisy,
intermediate-scale quantum (NISC) computers is therefore of high importance. Here,
we suggest the application of the so-called quantum state matching protocol for
testing purposes. This protocol was originally proposed to determine if an unknown
quantum state falls in a prescribed neighborhood of a reference state. We decompose
the unitary specific to the protocol and construct the quantum circuit implementing
one step of the dynamics for different characteristic parameters of the scheme and
present test results for two different IBM quantum computers. By comparing the ex-
perimentally obtained relative frequencies of success to the ideal success probability
with a maximum statistical tolerance, we discriminate statistical errors from device
specific ones. For the characterization of noise, we also use the fact that while the
output of the ideal protocol is insensitive to the internal phase of the input state,
the actual implementation may lead to deviations. For systematically varied inputs
we find that the device with the smaller quantum volume performs better on our
tests than the one with larger quantum volume, while for random inputs they show
a more similar performance.
∗Electronic address: ortega.adrian@wigner.hu
†Electronic address: kalman.orsolya@wigner.hu
‡Electronic address: kiss.tamas@wigner.hu
arXiv:2210.09674v2 [quant-ph] 14 Feb 2023
2
I. INTRODUCTION
The fields of Quantum Computation and Quantum Information have received a huge boost
in the last years with the advent of “public” quantum computation. Current devices can
be accessed remotely, opening the possibility for the larger public to carry out experiments
and to test them by running programs. Quantum computers (qcs) can be based on several
different physical systems such as superconducting qubits [1–3], trapped ions [4], photonic
devices [5] and neutral atoms [6]. Given all these possibilities, questions, such as compu-
tational efficiency, error correction capability, stability and computational power start to
become important matters for future applications.
In order to discriminate among the different technologies or to decide the optimal domain of
applicability of a given quantum computer one needs to devise “measure sticks” or bench-
marks. In the current so-called Noisy Intermediate-Scale Quantum (NISQ) era [7], the
question of what a suitable benchmark is, becomes tricky because we are still dealing with
“unfinished” technologies: qcs that contain a lot of errors, do not support efficient error cor-
rection, have low computational power, among other missing traits that classical computers
have already overcome [8]. Indeed, an argument has been made about the current field of
quantum computer benchmarking, stressing the point that we are still in the exploratory
stage [9]. The last few years have brought the arrival of the first quantum benchmarks, the
most prominent ones being the Quantum Volume (QV) [10, 11] and the Q-score [12]. Yet,
the field is only starting and there is still a long road ahead.
Current quantum benchmarks can be divided roughly into two categories (not taking into
account benchmarks related to temporal stability [13]): the first is based on randomized
circuits such as randomized benchmarking [14–17], the quantum volume [11], or random
circuits with a certain (mirror) structure [18]; the second is based on the successful achieve-
ment of certain hallmark protocols, such as the Bernstein-Vazirani algorithm and Grover’s
search [19–21], the Bell test, the matrix inversion procedure or Schr¨odinger’s microscope [22],
as well as algorithms used in quantum chemistry [23]. Benchmarks based on randomized
circuits in which a probability distribution is sampled are in general a good starting point to
test qcs, but they usually average out particular errors [22]. These particular errors might
become important, especially if the given quantum computer is used to perform a specific
task.
3
Our work follows the approach of the second category, as it is based on the so-called quantum
state matching protocol [24]. In a single step of this protocol a specific entangling operation
is applied on a pair of qubits, both prepared in the same initial state. Then one member
of the pair is measured and depending on the result of the measurement, the other member
of the pair is post-selected or discarded. This procedure leads to a complex nonlinear
transformation on the post-selected qubit. Note that a similar procedure is applied to realize
the Schr¨odinger microscope [22]. The transformation in our case is constructed in a way
that it has two superattractive fixed points [25] (which correspond to orthogonal quantum
states), with their respective basins of attraction being separated by a circle on the Bloch
sphere. By iterating the protocol one can decide whether a given unknown quantum state
falls in one of the basins of attraction, i.e., whether it is in the circle-shaped neighborhood
of one of the superattractive states. The radius of the circle can be prescribed, in a given
implementation of the scheme it determines the matrix elements of the entangling unitary,
and also affects the probability of success of the protocol.
In this paper, we show how one can employ this protocol to test qcs. In contrast to the
nonlinear protocol realizing the Schr¨odinger microscope [22], where the dynamics does not
possess any attractive cycles, thus all initial states are chaotic, in our scheme, the nonlinear
transformation has superattractive fixed points and a tuneable success probability. In this
way, the protocol itself can decrease initial noise, while such fluctuations may be enhanced
in the case of the Schr¨odinger microscope [26].
The paper is organised as follows. Section II describes the ideal protocol and introduces
the specific entangling unitary that is involved in it. Then, in Sec. III we determine the
optimal decomposition of this unitary into a minimum number of programmable quantum
gates. In Sec. IV we describe the statistical framework used to test the quantum computers,
while in Sec. V we present and analyse the results obtained from real qcs. We compare
results obtained by using systematically varied inputs (Sec. V A) and randomly chosen inputs
(Sec. V B), as well as results post-processed with readout error mitigation (Sec. V C). In
Section VI we conclude and give an outlook on future directions. Appendix A presents the
dates of the experiments discussed in the paper.
4
II. THE IDEAL PROTOCOL
Errors in current NISQ computers lead to deviations from the desired pure output state of
a quantum computation. These errors can be systematic, which do not necessarily change
the purity of the qubit state, or random ones, leading to mixed outputs. One might wish
to be able to decide whether such an output state is “close enough” to a desired pure state.
The quantum state matching protocol was originally proposed for this task [24]: given a
pure qubit state as a reference and a circular neighborhood around it, one can design a
scheme which transforms the unknown state closer to the reference state if it was originally
inside the prescribed circle, or otherwise to a state that is orthogonal to the reference state.
Assuming that the unknown state is at hand in many copies, the scheme can be further
iterated, and due to the superattractive nature of the transformation it can, after a few
steps, match the unknown state to the reference or its orthogonal pair, which can then be
discriminated.
The fast convergence of the above mentioned protocol is due to the nonlinear nature of
the underlying quantum state transformation. The nonlinearity arises because one takes
two copies of the unknown state |Φ0i, then applies a specific entangling unitary (which is
determined by the reference state and the radius of tolerance) and then measures one of
the qubits. If the measurement result is 0, then the other qubit undergoes a nonlinear
transformation compared to its initial state. It has been proven in Ref. [24] that one can
think of the entangling unitary as being composed of local rotations, which are determined
by the position of the reference state on the Bloch sphere, and a two-qubit unitary, which is
determined by the prescribed tolerance. A scheme containing solely this two-qubit unitary
realizes quantum state matching to the reference state |0i. In this work, we will focus on
the implementation of this latter protocol, which we describe in what follows.
Mathematically, a pure state of a two-level system can be written as
|Φ0i=N0(|0i+z|1i),(1)
where z∈C∪ ∞ and N0is a normalization factor. |Φ0ican be represented as a point on
the Bloch (or the Riemann) sphere or, equivalently, it can be represented as a point on the
complex plane, the two representations being related by the stereographic projection. Let us
denote by the radius of the circular shaped tolerance region on the complex plane around
5
the origin (representing the quantum state |0i). Then, let us take two qubits, both prepared
in the same intial state |Φ0i, and apply the entangling unitary
U=
−1
√2√1−21
√2√1−20
01
√2
1
√20
0 0 0 1
√1−21
√2−1
√20
,where 0 <  ≤1.(2)
It can be easily seen that if one measures the second qubit to be in state |0i, then the state
of the first qubit can be written as
|Φ1i=N1(|0i+f(z)|1i),where f(z) = z2
.(3)
Note that this is a quadratic (nonlinear) transformation of the initial state |Φ0i, represented
by the complex number z. If one iterates this protocol, then initial states with |z|<  will
converge to |0i, while those with |z|>  will converge to |1i, as the f(z) complex map has
two (super)attractive fixed points: 0 and ∞(corresponing to the quantum states |0iand
|1i, respectively). The two regions of convergence (the so-called Fatou set) are separated
by a circle (the so-called Julia set) of radius containing points which do not converge, but
evolve chaotically [25], [27]. We note that superattractivity of the quantum states |0iand
|1iis advantageous for the protocol, as it ensures the fastest possible convergence, yet, the
closer the initial unknown state is to the -circle, the more iterations are needed for the
initial state to be matched with the reference state (with a given accuracy).
In a quantum circuit the iteration of the protocol (and the corresponding complex map
f(z)) can be carried out as depicted in Figure 1. In order to carry out niterations, one
needs 2nqubits, all prepared in the same initial state |Φ0i. For the first iteration, one forms
2n−1pairs of these qubits and applies on every pair a Ugate. Every subsequent iterational
step requires 2 times less Ugates compared to the previous step, so that in the complete
circuit one needs to apply Pn
j=1 2j−1= 2n−1 times the Ugate. Note that in the original
proposal [24], one needs to perform measurements on one member of each pair and then
post-select only the successfully transformed qubits in each step. As this so-called “mid-
circuit” measurement is not yet implemented in current commercially available quantum
computers, one needs to postpone the post-selection step to the end of the quantum circuit,
where all qubits are measured at once.
摘要：

TestingquantumcomputerswiththeprotocolofquantumstatematchingAdrianOrtega,1,OrsolyaKalman,1,yandTamasKiss1,z1WignerRCP,Konkoly-ThegeM.u.29-33,H-1121Budapest,Hungary(Dated:February15,2023)Thepresenceofnoiseinquantumcomputershinderstheire ectiveoperation.Eventhoughquantumerrorcorrectioncantheoretic...

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