11institutetext:  ICUBE, Université de Strasbourg 22institutetext: ISIS,
Université de Strasbourg 33institutetext: QuantFi, Paris

# Evolutionary reduction of the laser noise impact on quantum gates

Tam’si Ley 11 3 3 0009-0001-5881-5689 Anna Ouskova Leonteva 11 Johannes
Schachenmayer 22 Pierre Collet 11

(April 2023)

###### Abstract

As the size of quantum hardware progressively increases, the conjectured
computational advantages of quantum technologies tend to be threatened by
noise, which randomly corrupts the design of quantum logical gates. Several
methods already exist to reduce the impacts of noise on that matter. However,
a reliable and user-friendly one to reduce the noise impact has not been
presented yet. Addressing this issue, this paper proposes a relevant method
based on evolutionary optimisation and modulation of the gate design. This
method consists of two parts : a model of quantum gate design with time-
dependent noise terms, parameterised by a vector of laser phases, and an
evolutionary optimisation platform aimed at satisfying a trade-off between the
gate fidelity and a pulse duration-related metric of the time consuming
simulation model. This feature is the main novelty of this work. Another
advantage is the ability to treat any noise spectrum, regardless of its
characteristics (e.g., variance, frequency range, etc). A thorough validation
of the method is presented, which is based on empirical averaging of random
gate trajectories. It is shown that evolutionary based method is successfully
applied for noise mitigation. It is expected that the proposed method will
help designing more and more noise-resisting quantum gates.

###### Keywords:

Quantum computing Noise resilience Optimisation

April 2023

##  1 Introduction

Quantum computing uses the laws of quantum physics to perform logical
operations inaccessible to classical means: e.g., superposition and
entanglement. In theory, an ideal quantum computer would solve some
mathematical problems with exponentially fewer computing resources than any
classical counterpart [1]. It applies to practical fields such as quantum
chemistry [2], materials science [3], and cryptography [4].

However, quantum systems are highly prone to a noise, coming from interactions
with the environment, or from imperfections in the control apparatus (e.g.,
laser, electronic devices). Due to analog nature of those quantum systems, the
noise critically affects the performance of quantum computers. Moreover, the
noise also dramatically reduces the quantum computational power, and would
prevent quantum computers from reaching any advantage on practical problems
with regard to the classical counterparts. This can be seen by the following
result: assuming a constant error rate and without any effective correction
strategy, a large variety of random quantum circuits (i.e programs) are
efficiently simulated by classical computers [5], [6], [7], which would be far
more difficult without noise. The ability to overcome noise is the central
question raised by quantum computing skeptics [17]. Indeed, quantum circuits
(programs) solving relevant problems, require millions of elementary quantum
gates (logical operators). Thus, even with small noise rates, they need
efficient strategies to suppress errors. The latter exist in theory, but they
require average gate infidelity much lower than what current noise levels
impose. Reducing the impact of noise on the gate design is a highly non-
trivial challenge, for which several solutions have been proposed: e.g., [14],
[15], [16]. Generally, these methods model the gate infidelity (inaccuracy)
with equations that require long simulations, and do not cope well with a high
standard deviation or high frequency fluctuations in the noise.
M Addressing these issues, in this work, we focus on adapting a flat laser
pulse suffering amplitude and dephasing noises, in order to implement a
Rydberg blockade two-qubit gate, which is specific to the cold-atoms hardware.
The theoretical information about that hardware is largely presented in the
literature: e.g., [20]. In order to reduce the noise and simultaneously, to
reduce the time spent in the Rydberg state, we formulate a quantum gate design
as a continuous non-constrained bi-objective problem based on a time-expensive
computer simulation. Taking into account the number of objectives ($2$) and
the large number of decision variables ($51$), we set 100 candidate solutions
in the population, where a computation time of around 1 minutes per each
solution. In order to reduce the computational complexity, we use the CPU-
parallel celebrated multi-objective algorithm Non-dominated Sorting Genetic
Algorithm III (NSGA-III) provided by the EASEA (EAsy Specification of
Evolutionary Algorithms) platform described in [19].
M To our best knowledge, this work is the first attempt to present a quantum
gate design task as a continuous large-scale multi-objective problem, solved
by parallel optimisation algorithm. Our method is not limited by any
characteristic of the noise, and allows us to select a pulse sequence (laser
phases) and duration to implement the target gate. Moreover, EASEA provides
different single and multi-objective optimisation algorithms, which can be
easily applied for the problem of laser noise reducing in the further
researches in order to investigate an impact of different optimisation
techniques on the performance in terms of the accuracy and speed up.
M This paper is organized as follows. Section 2 briefly outlines the related
works. Section 3 describes the problem definition. Section 4 presents the
proposed method based on the evolutionary optimisation. The experiments and
results are shown in Section 5. Section 6 concludes this article.

##  2 Related works

M In this section, we briefly present the related works addressing the problem
of laser noise impact reduction for quantum gate design presented in the
literature. They can be broadly divided in four categories.

The first one is called noise mitigation, and consists of repeating some gates
throughout the circuit, in order to reduce the influence of noise on the
computation result [8]. This technique generally assumes that the gate noise
is uncorrelated in time. However, correlated noise has been shown to bypass
the current noise mitigation strategies, as experimentally proved in [9].
Thus, non-trivial correlations in time must be accounted for, when analysing
noise impact on quantum computation.

The second category is robust optimal control, as was outlined in [10]. It
converts noise impact minimisation, into an optimal control problem in an
extended space with appropriate boundary conditions, and derives semi-
analytical solutions to that problem. However, it only treats time-independent
noise in a single direction (e.g amplitude or dephasing). Adapting that
technique to time-dependent noise would increase the problem dimension
exponentially in the number of harmonics, when considering all the cross
products between operators. A slight variation of this approach is "inverse
geometric optimisation", as introduced in [11]. The idea is to convert noise
impact reduction into a Euler-Lagrange problem, leading to a second order non-
linear differential equation. This approach allows to directly track the
trajectory of the quantum state in a semi-analytical formulation. However,
this method cannot be currently be adapted to either dephasing and/or time-
dependent noise, due do its mathematical formulation [11].

The third category is the filter function formalism, as outlined in [14] [15]
[16]. The idea is to formulate the average gate infidelity as a function of
both the laser controls and the noise power spectral density. Then, it is
optimised with differentiable methods such as gradient descent. However, that
method fails when the noise intensity is of order $10\%$ of the laser pulse
intensity, such that the implicit first-order truncation does not ensure a low
enough infidelity.

The fourth category consists of estimating the noise impact with Monte Carlo
summation of random gate trajectories; and to optimise the laser pulse with a
separate algorithm. In [13], the laser control is decomposed in Fourier
series, and the Fourier coefficients are optimized through the Nelder-Mead
algorithm. In their work, dephasing noise is considered relative, like
amplitude noise. This stands in contradiction to the convention in the
literature. Our present work follows a similar path, but characterises the
laser control with flat pulses, whose phases are optimised by a multi-
objective algorithm.

##  3 Problem definition

###  3.1 Context

Optically-trapped cold atoms constitute a promising hardware architecture for
long term quantum computing due to long qubit life time, maturity of laser
technology and flexibility of two-qubit interactions [20]. The idea of this
technology is to encode bits as two specific orbitals, called "0" and "1",
populated by the valence electron of some chosen special atomic species [20].

Figure 1: Atomic transitions and Rydberg interaction. Illustration adapted
from [21].

Upon interaction with a nearby electromagnetic field, electrons switch between
orbitals. This is usually done with laser tuned in resonance with the atomic
transition frequency between the said orbitals. In the literature, a
$\pi$-pulse refers to a complete transfer between two orbitals, with a
multiplication by the imaginary unit of the relevant orbital coordinate in the
quantum state vector. Rydberg states refer to orbitals far away from the
nucleus, exhibiting high electric dipole moments. When the resulting electric
potential is much larger than the power of the electromagnetic field, the
Rydberg blockade prevents two electrons to be simultaneously in a Rydberg
state. This happens when atoms are closed to each other. The protocol outlined
in [18] allows to design a universal two-qubit gate, using only the Rydberg
blockade and four $\pi$-pulses between the "1" and a Rydberg orbitals. Thus,
as we focus on laser noise, we only have to optimise the $\pi$-pulse. Figure 1
shows a simplified view of the atomic transition described above, where $r$ is
a Rydberg state, $B$ is the Rydberg blockade strength and $\Omega_{j}$ is a
function characterising the laser pulse.

###  3.2 Mathematical model

Due to the mathematical description of quantum mechanics, the electron’s
quantum state is represented as a complex unit vector, whose dimension is the
number of considered orbitals. In our case, there are 2 : the $1$ and $r$
orbitals, as $0$ is excluded from the transitions. Since the evolution of the
vector is linear and norm-preserving, it is represented as a $2\times 2$
complex unitary matrix $U(t)$, with $U(0)$ being the identity matrix. The
interaction between the atom and an electromagnetic field, inducing the
orbital transition, is mathematically represented as the following
differential equation:

$\dot{U}(t)=\imath\hat{H}(t)U(t)$ (1)

where $\hat{H}(t)$ is the hamiltonian matrix, which corresponds to the energy
configuration of the system. The transition between two atomic orbitals is
characterised by the complex Rabi frequency $\Omega$, as shown in the equation
below:

$\hat{H}(t)=\begin{pmatrix}\varepsilon_{d}(t)&(1+\varepsilon_{a}(t))\Omega^{*}(t)\\\
(1+\varepsilon_{a}(t))\Omega(t)&-\varepsilon_{d}(t)\end{pmatrix}$ (2)

where $\varepsilon_{d}$ is the dephasing noise, $\varepsilon_{a}$ is the
amplitude noise. In the absence of noise, the pulse will be the fastest, if
the Rabi frequency has a constant modulus, as explained in [21]. Since we want
the gates to be fastest as possible, in order to avoid other types of noises,
we accordingly parameterise the Rabi frequency :

$\Omega=2\pi f_{\max}e^{\imath\varphi}$ (3)

where $f_{\max}$ is the maximal frequency of the laser pulses and $\varphi$ is
the phases of the laser pulses. In the noiseless case, the phase is constant
$\varphi=0$. Here, we set it as a piece-wise constant function, with a fixed
number of slices ($N$): $\varphi(t)=\varphi_{i}$ for
$t\in[\frac{iT}{N},\frac{(i+1)T}{N}]$. The solution of the differential
Equation 1 is a random matrix written as $U(\varphi,t)$. For the sake of
clarity, the list of all the mathematical symbols used in this work is
summarised in Table 1.

Symbol | Significance (unit)
---|---
$\ket{0}$ | Ground orbital (–)
$\ket{1}$ | Excited orbital (–)
$\ket{r}$ | Rydberg orbital (–)
$\imath$ | Complex imaginary unit (–)
$t$ | Running time (1/$f_{\max}$)
$T$ | Pulse duration (1/$f_{\max}$)
$U$ | Pulse matrix (–)
$\dot{U}$ | Derivative of pulse matrix (–)
$\Omega$ | Rabi frequency (rad $f_{\max}$)
$\Omega^{*}$ | Complex conjugate thereof (rad $f_{\max}$)
$\hat{H}$ | Hamiltonian matrix (rad $f_{\max}$)
$U_{0}$ | Target pulse matrix (–)
$f_{\max}$ | Maximal Rabi frequency (1)
$\nu_{0}$ | Maximal noise frequency ($f_{\max}$)
$\boldsymbol{\varphi}$ | Vector of laser phases (rad)
$N$ | Number of time slices/laser phases
$F$ | Pulse infidelity (-)
$G$ | Time in Rydberg state (1/$f_{\max}$)
$\varepsilon_{a}$ | Amplitude noise (rad $f_{\max}$)
$\varepsilon_{d}$ | Detuning noise (rad $f_{\max}$)
Table 1: List of symbols used in the problem definition

This model can be transformed to an optimisation problem. From an optimization
point of view, the real-world quantum gate design problem, presented above, is
a typical expensive continuous bi-objective optimization problem without
constraints. A bi-objective problem aims at optimizing two values of interest
(objectives), which are antagonist. The answer is a set of trade-offs
solutions (a vector with the input parameters) between the defined objectives.
In section 3.3, we present its inputs and objectives.

###  3.3 Values of interest and input parameters

In this study, we consider 2 values of interests (objective functions) and 51
input parameters.

####  3.3.1 Input parameters:

The parameters and their ranges of variations are summed up in Table 2,

Table 2: List of all inputs Input | Min | Max | Unit
---|---|---|---
Phases: $\boldsymbol{\varphi}[0-50]$ | 0 | 2$\pi$ | rad
Time: $T$ | 1 | 5 | $1/f_{\max}$

where the following notation is used:

  * •

$\boldsymbol{\varphi}$ is a vector of $N$ elements, which refers to the values
of the laser phase in the orbital transitions. In the current model $N=50$ as
a reasonable number required for both processes: simulation and optimisation.

  * •

$T$ is the duration of the model simulation, which has an impact on the time
spent by the electron in the Rydberg state ($G$);

  * •

$f_{\max}$ is the maximal frequency the laser emits to. In our problem
definition, we normalise all units with $f_{\max}=1$, which means that the
model can be applied to any value of that frequency. In recent experiments,
typically $f_{\max}=1$ GHz.

####  3.3.2 Objective functions

We seek to make $U(\boldsymbol{\varphi},T)$ (see Equation 1) as close as
possible to $U_{0}:=\begin{pmatrix}0&\imath\\\ \imath&0\end{pmatrix}$, because
this matrix represents the $\pi$-pulse that we look for, since we base our
work on the protocol of [18], which features only $\pi$-pulses. This motivates
us to define the gate infidelity as follows :

$F(\boldsymbol{\varphi},T):=\mathbb{E}(1-\left|\text{Tr}(U(\boldsymbol{\varphi},T)U_{0}^{\dagger})\right|^{2}/4)$
(4)

A important source of errors in quantum computation, is spontaneous photon
emission. In this random process, super excited states like the Rydberg ones,
tend to spontaneously revert to a lower-energy state, releasing a photon. This
cannot be prevented, so the only way to minimise the risk of occurrence is to
minimise the time spent by the electron in the Rydberg orbital. Assuming the
electron starts in the $1$ orbital, this quantity is given by :

$G(\boldsymbol{\varphi},T):=\mathbb{E}\left(\int_{0}^{T}|\langle
r|U(\boldsymbol{\varphi},t)|1\rangle|^{2}dt\right)$ (5)

Let’s briefly show that these two quality criteria are antagonistic.
Simulating flat pulses leads to the appearance of noise terms of the form
$\varepsilon(\frac{jT}{N})$ in the computations. Since the number $N$ of
slices is fixed, these terms tend to vanish as $T$ grows to infinity. Thus,
the impact of noise can be mitigated by extending the pulse duration, and
choosing a trivial $\boldsymbol{\varphi}$. In other words, increasing $T$
tends to reduce $F$, as well as simplifying the choice of
$\boldsymbol{\varphi}$, but it would potentially imply a larger value of $G$
and thus a higher risk of Rydberg state decay. On the other hand, $0\leq G\leq
T$ (because the integrand is lower than $1$), so an optimal $G$ would rather
be found by decreasing $T$, but in turn this complexifies the search for
optimal phases. Consequently, to find the trade-off between these objectives
is required.

##  4 Proposed Method

The goal of this study is to reduce the gate infidelity (i.e the impact of
laser noise) by finding the optimal laser phases $\boldsymbol{\varphi}$,
without increasing the time spent in the Rydberg state, in order to minimise
the risk of Rydberg state decay. Consequently, it will improve the global gate
quality. To achieve this goal, we propose a CPU-parallel multi-objective
optimisation-based method, since the quantum gate design can be formulated as
a continuous bi-objective computationally expensive problem (as was shown
above). Indeed, a multi-objective optimization process allows us to find: (i)
all possible non dominated solutions – combinations of $\boldsymbol{\varphi}$
and $T$, which is called the Pareto set; (ii) its image in the objective
space, which presents the Pareto front. Thus, we consider to define a set of
optimal parameters of the model (i.e., the phases vector of the laser pulses
and the duration of the model simulation), by minimizing the objective
functions (i.e., overall infidelity and time spent in the Rydberg state),
calculated by simulation.

Figure 2: Simplified scheme of the proposed method.

The proposed method consists of two following parts:

  * •

A Quantum gate design presented in Section 3.2, which consists of simulating
the laser pulse hamiltonian, on noisy trajectories; and computing the
objective functions with Monte Carlo average. For the sake of completeness,
the noise generation details are provided in online supplementary materials:
https://git.unistra.fr/tsley/thesis.

  * •

The EASEA platform, an open-source compiler, which supports single and multi-
objective optimization thanks to its own base of the templates of different
state-of-the-art algorithms [19]. The main advantage of EASEA is that it
allows scientists to apply evolutionary algorithms to solve their real-world
problems through a user-friendly interface with CPU and/or GPU
parallelisation. More precisely, this compiler automatically couples one of
the algorithm template files and a problem to be optimised, and provides the
C++ based code of the selected algorithm with the integrated problem in it and
the compiling file. Consequently, it suits perfectly as a software support for
our study. Notably, the quantum gate design model can be modified without
changing other parties of the code, due to the very flexible structure of
EASEA.

By this concept, we had to select an optimisation algorithm template (*.tpl
file), define the problem in terms of the pseudo-code (*.ez file) and define
the noise parameters, such as the spectrum (white, pink, etc.) and its level
w.r.t. maximal intensity of the laser pulse. The simplified scheme of the
considered method is schematically depicted in Figure 2. It details the
programming relation between: (i) the EASEA CPU-parallel evaluator with our
quantum gate design model presented on the left and (ii) the selected multi-
objective algorithm: Non-dominated Sorting Genetic Algorithm (NSGA-III), shown
on the right in Figure 2. NSGA-III originally presented in [22], is based on
the non-dominated sorting selection mechanism and uses a predefined set of
reference points to ensure diversity in the solutions [22]. NSGA-III was
selected in this work between other templates, because of the following
reasons: (i) it is efficient on several multi- and many-objective problems
[23] (in future, the current problem can be extended to a many-objective
version); (ii) it does not require any additional hyper-parameters; (iii) it
introduces a computationally fast approach, by which the reference point set
is adaptively updated on the fly based on the association status of each
reference point over a number of generations [22], [23]; (iv) it can handle
constraints without introducing any new parameter, which can be useful in our
next research, where some constraints will be taken into account; (v) it can
easily be scaled for solving single-objective problems [23]. The following
steps explain how this method performs:

  1. 1.

The selected optimization algorithm, e.g., NSGA-III, starts by randomly
generating the initial set of solutions, where each solution consists of the
parameters presented in Table 2;

  2. 2.

Its main cycle is repeatedly executed until the last generation (see the right
part of Figure 2);

  3. 3.

NSGA-III creates the new child population $Q_{t}$ from the parent population
$P_{t}$ by the classical breeding operators used in NSGA-III [22]: Simulated
Binary crossover (SBX) crossover and Polynomial mutation operator.

  4. 4.

The EASEA Evaluator executes 100 threads to compute $F$ and $G$ in parallel
(see the left part of Figure 2). Each executed thread works out the model code
with its own combination of input parameters and generated noise.

  5. 5.

Each thread returns the output values (the overall infidelity ($F$) and the
time spent in the Rydberg state ($G$)) back to the NSGA-III.

  6. 6.

NSGA-III performs according to its description presented in [22] and goes to
the next generation.

Also, important to emphasise that NSGA-III of EASEA is designed for CPU
parallel use (several hundred computing threads), which allows us to reduce
the runtime.

##  5 Experimental analysis

###  5.1 Experimental design

All values of the setting parameters (both for the simulation model and for
the optimisation algorithm) presented below, are the same during all provided
experiments. For all experiments we use the problem definition presented in
Section 3.3. The decision variables are defined as the presented input
parameters. For single-objective study, we use only $F$ as a single
optimisation function and applied the well-known Covariance Matrix Adaptation
Evolution Strategy (CMA-ES) algorithm [25] from the EASEA plateform. For
multi-objective study, as it was presented above, the NSGA-III algorithm is
applied.

####  5.1.1 Model Settings

In this paper, we chose to work with $1/f$ noise, also known as pink or
flicker noise, since it is widely present in electronic devices, and thus in
the quantum hardware that they control. So, we generate $10^{6}$ samples
thereof, each time as a sum of sines and cosines with $25$ uniformly log-
spaced harmonics, in the interval $[f_{\max},100f_{\max}]$, thereby
approximating a high-frequency noise. Remind that $f_{\max}$ is the
characteristic unit of the problem, and is set to $1$. Remind that the
optimisation target is a $\pi$-pulse to implement a Rydberg blockade two-qubit
CZ gate. We effectuate three tests with respective noise levels of 10%, 20%
and 30% of the maximal laser pulse intensity.

####  5.1.2 Optimisation Settings

For all experiments, the number of decision variables equals 51, according to
the problem definition. The parameter settings of CMA-ES is limited by the
number of generations = 300, the population size = 100 and the number of
threads = 100.

Table 3:  Parameter settings of NSGA-III Parameter | $p_{c}$ | $\eta_{c}$ | $p_{m}$ | $\eta_{m}$ | $M$ | $P$ | $H$ | $N$ | $G$ | $Th$
---|---|---|---|---|---|---|---|---|---|---
Value | 1.0 | 30 | 1/51 | 20 | 2 | 99 | 100 | 100 | 200 | 100

The parameter settings of NSGA-III are presented in Table 3, where $p_{c}$ is
Simulated Binary (SBX) crossover probability, $\eta_{c}$ is crossover
distribution index, $p_{m}$ is polynomial mutation probability, $\eta_{m}$ is
polynomial distribution index, $M$ is the number of objectives, $P$ is a given
integer value, which refers to the number of divisions considered along of
each objective axis, $H$ is the number of reference points, $N$ is the
population size, $G$ is the number of generation and $Th$ is the number of
threads. The selection of $P$, $H$ and $N$ was made according to the rules
defined in [22]: (i) if a problem has less number of objectives, a larger
number of $P$ is required (100 is recommended value in bi-objective case);
(ii) $H$ is calculated by the approach of Das and Dennis [24]; (iii) the
population size is set as the smallest multiple of four higher than the number
of reference points (in our case $N$=$H$).

###  5.2 Result analysis

The obtained results are shown in Figure 3 and described below.

|
---|---
a) Best solution | b) Pareto Front
Figure 3: Obtained results.

####  5.2.1 Single-objective

This study started with a single-objective optimisation of $F$, with a fixed
$T=1$ and a noise intensity of 10%. After 136 generations of 100 candidates,
the algorithm CMA-ES found an optimal solution $F=0.005052$. This solution in
terms of the coordinates of $\boldsymbol{\varphi}$ is presented in log scale
in Figure 3 (a). This result is better than the gate infidelity $F=0.0055$
obtained with the naive choice of setting all the 50 phase to be zero:
$\boldsymbol{\varphi}=\mathbb{0}$. This result confirmed us that artificial
evolution can bring some improvements in robust gate designs, and motivated us
to explore multi-criteria optimisation for simultaneous antagonistic problems.

####  5.2.2 Multi-objective

The Pareto fronts obtained for three selected noise levels (%10, %20 and %30
of $2\pi f_{\max}$) are presented in Figure 3 (b), where the axe $F$ is
presented in log scale. As it seen from Figure 3 (b), the lowest minimum of
$F$ was obtained in the case of 10% noise level (presented in blue).

For the rightmost element of the Pareto front, the pulses we find result in a
compromise between an infidelity slightly higher than the naive case, and a
shorter time spent in the Rydberg state, signalling a preference in reducing
Rydberg decay risk. For elements of the leftmost Pareto front, the situation
is reversed. This shows that NSGA-III drives the solutions to satisfactory
compromises, which should be investigated w.r.t. possible values of the
Rydberg decay rate. For all cases, the shape of the Pareto Front is slightly
discontinuous, however the solutions are not irregularly distributed in the
objective space. It shows that for all selected noise levels, NSGA-III is able
to find solutions in the different regions of the objective space, providing
the possible variations of trade-off between the overall infidelity and the
time spent in the Rydberg state. We assume that these results can be improved
by increasing the population size and the number of generations.

##  6 Conclusion

The multi-objective based method we propose aims at introducing a reliable
gate design, which allows us to reduce both the laser noise impact and the
risk of Rydberg state decay. As a result, this method provides a variety of
trade-offs between these two objectives, within a reasonable run-time, thanks
to CPU-parallelization. It can be easily applied to different noise and laser
pulses models. The problem studied here focuses on decomposing a flat pulse
affected with pink noise on both amplitude and phase with different noise
intensity levels. With the trust of the multi-objective optimization method,
thanks to EASEA platform, it is now possible to look for interesting gate
design features for a wide variety of noise with a reduced run-time. Moreover,
as the EASEA platform provides different single and multi-objective
optimisation algorithms, which can be easily coupled with the problem of laser
noise reduction, we prospect to investigate the impact of different
optimisation techniques on the performance in terms of the accuracy and speed
up in further works. Also, further research with different noise types would
be considered. However, our work inherited the drawbacks of the protocol
outlined in [18], in the sense that its setup would require an individual
laser for each atom, which is difficult to attain. Therefore, beyond simply
noise-corrected $\pi$-pulses, future works will involve a global pulse model
with higher degrees of freedom, similarly to [21]. This would correspond to a
single laser for all the atoms, able to design multi-qubit Rydberg blockade
gates. It is expected that this method can help to make a qualitative step
toward developing robust quantum gates.

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##  7 Supplementary material

###  7.1 Symbols table

Symbol | Significance | Dimension (Unit)
---|---|---
$\ket{0}$ | Ground orbital | Dimensionless
$\ket{1}$ | Excited orbital | Dimensionless
$\ket{r}$ | Rydberg orbital | Dimensionless
$\imath$ | Complex imaginary unit | Dimensionless
$t$ | Running time | Time (1/$f_{\max}$)
$T$ | Pulse duration | Time (1/$f_{\max}$)
$U$ | Pulse matrix | Dimensionless
$\dot{U}$ | Derivative of pulse matrix | Dimensionless
$\Omega$ | Rabi frequency | Angle frequency (rad $\cdot$ Hz)
$\Omega^{*}$ | Complex conjugate of the latter | Angle frequency (rad $\cdot$ Hz)
$\hat{H}$ | Hamiltonian matrix | Dimensionless
$U_{0}$ | Target pulse matrix | Dimensionless
$\nu_{0}$ | Maximal value of the Rabi frequency | Inverse time (Hz)
$f_{\max}$ | Maximal frequency reached by the noise | Inverse time (Hz)
$\varphi$ | Laser phase | Angle (rad)
$F$ | Pulse infidelity | Dimensionless
$G$ | Time in Rydberg state | Time (1/$f_{\max}$)
$\varepsilon_{a}$ | Amplitude noise | Angle frequency (rad Hz)
$\varepsilon_{d}$ | Detuning noise | Angle frequency (rad Hz)
Table 4: List of symbols

###  7.2 Noise generation

Pink noise is also referred to as "$1/f$ noise", since its power spectral
distribution is roughly the inverse function, though with a minimal and
maximal frequency cutoffs, since the inverse function is not integrable. Being
also named "flicker" noise, it is widely present in electronics, including the
ones that control quantum devices. Existing methods to compute pink noise with
trigonometric series present a lot of limitations : for example they require a
large number of harmonics to ensure convergence to the desired spectrum, and
thus are not presented explicitly in the literature. The following method
accurately computes pink Noise with a limited number of terms. At the best of
our knowledge, it has not been presented elsewhere.

$C(h)=\int_{0}^{\log(\nu_{\max})}\cos(2\pi
e^{x}h)dx=\log(\nu_{0})\int_{0}^{1}\cos(2\pi\nu_{0}^{s}h)ds$

Indeed, this integral can be approximated up to first order as :

$\frac{C(h)}{C(0)}=C_{M}(h)+O\left(\frac{1}{M}\right)$

Where

$\quad C_{M}(h):=\frac{1}{M}\sum_{n=0}^{M-1}\cos(2\pi\nu_{0}^{\frac{n}{M}}h)$

Thus, if $(a_{n},b_{n})$ are two i.i.d sequences of standard normal random
variables, then the expression

$X(t):=\frac{1}{M}\sum_{n=0}^{M-1}\cos(2\pi\nu_{\max}^{n/M}t)a_{n}+\sin(2\pi\nu_{\max}^{n/M}t)b_{n}$

defines a Gaussian process such that $\mathbb{E}(X(t+h)X(t))=C_{M}(h)$,
thereby providing a good approximation to pink noise.

###  7.3 Optimal coordinates of $\varphi$

The phase vector found by the single-objective optimisation mentioned in
subsection 5.2 is :

$\displaystyle\varphi=($ $\displaystyle
3.1117210566,3.0681531508,3.1062684923,3.1204162648,3.1511899718,$
$\displaystyle
2.9093600733,3.0648268235,3.0531992808,3.1219826294,3.0848634591,$
$\displaystyle
3.0181634846,3.0485478461,3.1157769830,3.1606018942,2.9632053356,$
$\displaystyle
3.0567039551,3.0493017295,3.1321501012,3.2415897692,3.1940974328,$
$\displaystyle
3.2861696576,3.2583179501,3.3060153192,3.2227544442,3.3887061970,$
$\displaystyle
3.2053472380,3.2594614180,3.2034174002,3.3582338303,3.1268169264,$
$\displaystyle
3.1433784957,3.2405135080,3.1008649385,3.2758823795,3.1663607416,$
$\displaystyle
3.1524414836,2.9931205862,3.1318277759,3.0787100614,3.0011544184,$
$\displaystyle
2.9331935381,3.0155597121,3.0218345718,2.9837460312,2.9705769473,$
$\displaystyle
3.0755616247,3.0868778008,3.0677822108,3.2140962589,3.1436841099$
$\displaystyle)$


# Twistor quadrics and black holes

Bernardo Araneda111Email<EMAIL_ADDRESS>
Max-Planck-Institut für Gravitationsphysik
(Albert-Einstein-Institut), Am Mühlenberg 1,
D-14476 Potsdam, Germany

###### Abstract

A simple procedure is given to construct curved, non-self-dual (complexified)
Kähler metrics on space-time in terms of deformations of holomorphic quadric
surfaces in flat twistor space. Imposing Lorentzian reality conditions, the
Schwarzschild, Kerr, and Plebański-Demiański space-times (among others) are
derived as examples of the construction.

##  1 Introduction

Twistor theory constitutes a remarkable approach to the description of the
complex structure of space-time [1, 2, 3, 4]. Motivated by the fact that the
celestial sphere of any event in Minkowski space has a natural complex
structure and is thus a Riemann sphere $\mathbb{CP}^{1}$ (cf. [5]), the space-
time manifold emerges as the moduli space of $\mathbb{CP}^{1}$’s (‘twistor
lines’) in twistor space. These lines are holomorphic, which implies that our
intuition about lines and their intersection in $\mathbb{R}^{3}$ continues to
hold in twistor space, and allows to define the conformal structure of space-
time by intersection of lines. Gravitation, in the twistor view, should
correspond to deformations of the flat twistor structure. Penrose’s non-linear
graviton construction [2] shows that this holds true for self-dual (or half-
flat) curved space-times.

If a geometry admits a twistor space and, in addition, it has some further
special structure, this extra structure is often holomorphically encoded in
the twistor space. For example, in Riemannian geometry, Pontecorvo showed [6]
that a Kähler metric in the conformal structure of a conformally half-flat
4-manifold corresponds to a holomorphic section of (the square root of) the
anti-canonical bundle of the twistor space, with two zeros on each twistor
line. This can also be understood in terms of a holomorphic surface that
intersects each twistor line at two points.

Self-dual curvature is a strong restriction for general relativity, as it
implies (conformally) flat space-time. In this note, inspired by Pontecorvo’s
construction and by the non-linear graviton, we describe a variation of the
twistor construction that produces non-self-dual (complexified) Kähler metrics
on space-time as deformations of holomorphic quadric surfaces in flat twistor
space. Part of the basic intuition is that two points define a line; this
carries over to twistor space since everything is holomorphic (the two points
in question being the intersection points of twistor lines with the quadric;
see below). Imposing Lorentzian reality conditions, we show that the general
Plebański-Demiański class of space-times [7, 8], which includes the standard
black hole metrics of general relativity such as Schwarzschild, Kerr, etc., is
recovered by this construction.

##  2 Twistor quadrics and Kähler metrics

###  2.1 Preliminaries

We start off by introducing some basic definitions to set our conventions. Let
$(M,g)$ be a 4-dimensional, orientable Riemannian or Lorentzian manifold. We
also allow complexified geometries. We say that a $(1,1)$ tensor $J$ is an
almost-complex structure if it satisfies $J^{2}=-\mathbb{I}$ and its $(\pm{\rm
i})$-eigenspaces $T^{\pm}$ have the same rank. We say that $J$ is, in
addition, compatible with $g$ if it holds $g(J\cdot,J\cdot)=g(\cdot,\cdot)$;
in this case we refer to $(g,J)$ as an almost-Hermitian structure. The
$(\pm{\rm i})$-eigenspaces $T^{\pm}$ of $J$ split the tangent bundle as
$T^{+}\oplus T^{-}$. We say that $J$ is integrable, and is thus a complex
structure, if $T^{\pm}$ are both involutive under the Lie bracket, i.e.
$[T^{\pm},T^{\pm}]\subset T^{\pm}$ (for both signs). If this is satisfied and
$J$ is also compatible with $g$, then $(g,J)$ is a Hermitian structure.

Given an almost-Hermitian structure, the fundamental 2-form is defined by
$\kappa(\cdot,\cdot)=g(J\cdot,\cdot)$. The 2-form $\kappa$ and $J$ are
compatible in the sense that $\kappa(J\cdot,J\cdot)=\kappa(\cdot,\cdot)$. Note
that any two of $g,J,\kappa$ determines the third; in particular, given
compatible $\kappa,J$, the metric $g$ is determined by
$g(\cdot,\cdot)=\kappa(\cdot,J\cdot)$. Finally, we say that the geometry is
(complexified) Kähler if $(g,J)$ is Hermitian and ${\rm d}\kappa=0$. In this
case, $\kappa$ is also referred to as the symplectic form.

Concerning reality conditions, one can show that given an almost-Hermitian
structure, the fundamental 2-form is an eigenform of the Hodge star operator,
so it is self-dual (SD) or anti-self-dual (ASD). In Riemann signature, (A)SD
2-forms are real, whereas in Lorentz signature they are complex. Therefore,
the tensor $J$ is real-valued in Riemann signature, and complex-valued in
Lorentz signature. Lorentzian Kähler geometry was thoroughly investigated by
Flaherty [9].

###  2.2 Twistor space

Let $\mathbb{CM}$ denote complexified Minkowski space, with complexified
inertial coordinates $t_{\rm c},x_{\rm c},y_{\rm c},z_{\rm c}$ and flat
holomorphic metric $\eta={\rm d}t_{\rm c}^{2}-{\rm d}x_{\rm c}^{2}-{\rm
d}y_{\rm c}^{2}-{\rm d}z_{\rm c}^{2}$. Introduce double null coordinates
$u_{\rm c},v_{\rm c},w_{\rm c},\tilde{w}_{\rm c}$ by $u_{\rm
c}=\frac{1}{\sqrt{2}}(t_{\rm c}+z_{\rm c})$, $v_{\rm
c}=\frac{1}{\sqrt{2}}(t_{\rm c}-z_{\rm c})$, $w_{\rm
c}=\frac{1}{\sqrt{2}}(x_{\rm c}+{\rm i}y_{\rm c})$, $\tilde{w}_{\rm
c}=\frac{1}{\sqrt{2}}(x_{\rm c}-{\rm i}y_{\rm c})$. Twistor space is the
manifold $\mathbb{PT}=\mathbb{CP}^{3}\backslash\mathbb{CP}^{1}$, and it is
related to space-time via the incidence relation

$\displaystyle\left(\begin{matrix}Z^{0}\\\ Z^{1}\end{matrix}\right)={\rm
i}\left(\begin{matrix}u_{\rm c}&w_{\rm c}\\\ \tilde{w}_{\rm c}&v_{\rm
c}\end{matrix}\right)\left(\begin{matrix}Z^{2}\\\ Z^{3}\end{matrix}\right)$
(1)

where we use homogeneous coordinates $Z^{\alpha}=(Z^{0},Z^{1},Z^{2},Z^{3})$ on
$\mathbb{CP}^{3}$ (the $\mathbb{CP}^{1}$ removed corresponding to
$Z^{2}=Z^{3}=0$). The twistor correspondence (1) is non-local. Fixing
$Z^{\alpha}\in\mathbb{PT}$, the set of space-time points satisfying (1) is a
totally null 2-surface in $\mathbb{CM}$, called ‘$\alpha$-surface’. Fixing
$(u_{\rm c},v_{\rm c},w_{\rm c},\tilde{w}_{\rm c})\in\mathbb{CM}$, the set of
$Z^{\alpha}$ satisfying (1) is a (holomorphic, linearly embedded) Riemann
sphere $L_{x}\cong\mathbb{CP}^{1}$, which is called ‘twistor line’. Space-time
is the moduli space of twistor lines in $\mathbb{PT}$, and twistor space is
the moduli space of $\alpha$-surfaces in $\mathbb{CM}$. Inhomogeneous local
coordinates for twistor space are given by the equations that define
$\alpha$-surfaces, that is (putting $\zeta=Z^{3}/Z^{2}$, in a region with
$Z^{2}\neq 0$)

$\displaystyle\omega^{0}=u_{\rm c}+\zeta w_{\rm
c},\qquad\omega^{1}=\tilde{w}_{\rm c}+\zeta v_{\rm c},\qquad\zeta.$ (2)

For fixed $(u_{\rm c},v_{\rm c},w_{\rm c},\tilde{w}_{\rm c})$ and variable
$\zeta$, these three quantities describe a twistor line in $\mathbb{PT}$. For
fixed $(\omega^{0},\omega^{1},\zeta)$ and variable $(u_{\rm c},v_{\rm
c},w_{\rm c},\tilde{w}_{\rm c})$, (2) describe an $\alpha$-surface in
$\mathbb{CM}$.

Twistor space is fibered over $\mathbb{CP}^{1}$, being the total space of the
fiber bundle $\mathcal{O}(1)\oplus\mathcal{O}(1)\to\mathbb{CP}^{1}$. Here,
$\mathcal{O}(-1)$ is the tautological line bundle over $\mathbb{CP}^{1}$, and
$\mathcal{O}(k)=\mathcal{O}(-1)^{*\otimes k}$ ($k>0$). The base of the
fibration has homogeneous coordinates $Z^{2},Z^{3}$, or inhomogeneous
coordinate $\zeta$, and the fibers have coordinates $Z^{0},Z^{1}$. Each fiber
has an $\mathcal{O}(2)$-valued symplectic structure $\mu={\rm
d}Z^{0}\wedge{\rm d}Z^{1}$. All of this is valid for flat space-times. One of
the main ideas in twistor theory is that gravitation, namely curved space-
times, should correspond to deformations of twistor structures. Penrose showed
[2] that this is true for half-flat space-times: he proved that an ASD, Ricci-
flat, complex space-time corresponds to a 3-dimensional complex manifold
$\mathcal{PT}$ obtained as a deformation of $\mathbb{PT}$ that preserves the
fibration $\mathcal{PT}\to\mathbb{CP}^{1}$ and the fiberwise symplectic
structure $\mu$.

###  2.3 Quadrics

In a Riemannian setting, twistor space can also be defined as the space of
almost-complex structures compatible with a 4-dimensional Riemannian conformal
structure. This space coincides with the (6-real-dimensional) projective spin
bundle, which can be shown to be a complex 3-manifold (the twistor space
$\mathcal{PT}$) if and only if the conformal structure is ASD.

In this context, one may ask what a Kähler metric in the conformal structure
corresponds to in twistor space. This was studied by Pontecorvo [6], who
showed that a (necessarily scalar-flat) Kähler metric corresponds to a
preferred (global, holomorphic) section of $K^{-1/2}_{\mathcal{PT}}$ which
vanishes at two points in each twistor line, where $K^{-1/2}_{\mathcal{PT}}$
is the square-root of the anti-canonical line bundle of $\mathcal{PT}$. We can
see this by first noticing that the bundle $K^{-1/2}_{\mathcal{PT}}$
restricted to a twistor line is $\mathcal{O}(2)$. Hitchin showed in [10,
Section 2] that the space $H^{0}(\mathcal{PT},\mathcal{O}(k))$ of global,
holomorphic sections of $\mathcal{O}(k)$ can be identified with $\ker(T_{k})$,
where $T_{k}$ is the valence-$k$ twistor operator. This implies that a section
$\chi$ of $K^{-1/2}_{\mathcal{PT}}$ corresponds to a valence-2 twistor-spinor,
or Killing spinor. The requirement that $\chi$ vanishes at two points in each
twistor line means that the Killing spinor is non-degenerate. Using then [11,
Lemma 2.1], this corresponds to a conformal Kähler structure on space-time.

In complexified flat space-time, we can also describe Pontecorvo’s
construction in terms of holomorphic quadric surfaces in flat twistor space
$\mathbb{PT}$. The space $H^{0}(\mathbb{CP}^{1},\mathcal{O}(k))$ consists of
degree $k$ homogeneous polynomials in $\mathbb{C}^{2}$. Thus, a section $\chi$
of $K^{-1/2}_{\mathbb{PT}}$, when restricted to a twistor line, is of the form

$\displaystyle\chi=A\zeta^{2}+2B\zeta+C$ (3)

for some $A,B,C$, where we are using an inhomogeneous coordinate $\zeta$ on
$\mathbb{CP}^{1}$. We can then think of $\chi$ as a holomorphic quadratic
function $\chi(Z^{\alpha})=Q_{\alpha\beta}Z^{\alpha}Z^{\beta}$ for some
symmetric $Q_{\alpha\beta}$. The expression (3) follows after using the
incidence relation (1), which also shows that $A=A(v_{\rm c},w_{\rm c})$,
$B=B(u_{\rm c},v_{\rm c},w_{\rm c},\tilde{w}_{\rm c})$ and $C=C(u_{\rm
c},\tilde{w}_{\rm c})$. The zero set of $\chi$ is a holomorphic quadric,
$\mathbb{Q}=\\{Z^{\alpha}\in\mathbb{PT}\>|\>\chi(Z^{\alpha})=0\\}$. From
Kerr’s theorem [1], the surface $\mathbb{Q}$ corresponds to a shear-free, null
geodesic congruence in $\mathbb{CM}$. The condition that $\chi$ vanishes at
two points in each twistor line $L_{x}$ is the same as saying that $L_{x}$
intersects $\mathbb{Q}$ at two points, corresponding to the two roots
$\zeta_{\pm}$ of the quadratic polynomial (3), that is
$\chi=A(\zeta-\zeta_{+})(\zeta-\zeta_{-})$, where
$\zeta_{\pm}=(-B\pm\sqrt{B^{2}-AC})/A$. We allow, however, the possibility of
twistor lines where the roots coincide; these correspond to caustics in the
ray congruence on space-time (and will later correspond to curvature
singularities for the curved, non-self-dual metric we will construct).
Importantly, the quadric is divided into two regions in twistor space:

$\displaystyle\mathbb{Q}=\mathbb{A}_{+}\cup\mathbb{A}_{-},$ (4)

where $\mathbb{A}_{\pm}$ can be described in local coordinates by any two of
(see (2))

$\displaystyle\omega^{0}_{\pm}=u_{\rm c}+\zeta_{\pm}w_{\rm
c},\qquad\omega^{1}_{\pm}=\tilde{w}_{\rm c}+\zeta_{\pm}v_{\rm
c},\qquad\zeta_{\pm},$ (5)

or by any function of them. For fixed $+$ or $-$, the three coordinates in (5)
are functionally dependent as a consequence of the quadric equation
$\chi(\omega^{0}_{\pm},\omega^{1}_{\pm},\zeta_{\pm})=0$. A simple example to
illustrate this (and to have in mind in general) is a product of planes, that
is, a quadric given by $Q_{\alpha\beta}=A^{+}_{(\alpha}A^{-}_{\beta)}$ for
some fixed $A^{\pm}_{\alpha}$. The two regions in (4) are in this case two
planes $\mathbb{A}_{\pm}=\\{Z^{\alpha}|A^{\pm}_{\alpha}Z^{\alpha}=0\\}$, and
the roots coincide in the twistor line corresponding to the intersection of
the planes, see e.g. [12, Fig. 6-11].

Since, generically, a twistor line $L_{x}$ intersects $\mathbb{Q}$ at two
points, and since two points define a unique line through them, the two
intersection points can also be used to characterise the twistor line $L_{x}$.
In other words, varying the line $L_{x}$, the intersection points serve as a
coordinate system on space-time. Given local holomorphic coordinates on
$\mathbb{A}_{\pm}$ (obtained e.g. from (5)), say $z^{A}_{\pm}$ with $A=0,1$,
the pair $(z^{A}_{+},z^{A}_{-})$ is the desired coordinate system on
$\mathbb{CM}$. The complex structure $J$ induced on space-time from the
quadric $\mathbb{Q}$ can then be shown to be

$\displaystyle J={\rm i}(\partial_{z^{A}_{+}}\otimes{\rm
d}z^{A}_{+}-\partial_{z^{A}_{-}}\otimes{\rm d}z^{A}_{-})$ (6)

where the Einstein summation convention is assumed. The tensor (6) is
compatible with the Minkowski metric. In particular, the vectors
$\partial_{z^{A}_{\pm}}$ are null. In fact, the construction so far is
conformally invariant. The Kähler structure is obtained from the symplectic
form, which can be shown to be

$\displaystyle\kappa=\frac{{\rm i}}{(B^{2}-AC)^{3/2}}[A{\rm d}u_{\rm
c}\wedge{\rm d}\tilde{w}_{\rm c}-B({\rm d}u_{\rm c}\wedge{\rm d}v_{\rm c}+{\rm
d}w_{\rm c}\wedge{\rm d}\tilde{w}_{\rm c})+C{\rm d}w_{\rm c}\wedge{\rm
d}v_{\rm c}]$ (7)

($A,B,C$ are defined in (3)). One can show this using, for example, the
Penrose transform for spin 1, with the twistor function
$f(Z^{\alpha})=[\chi(Z^{\alpha})]^{-2}$. In terms of quadric coordinates
$z^{A}_{\pm}$, the symplectic form (7) is

$\displaystyle\kappa=\kappa_{A\tilde{B}}{\rm d}z^{A}_{+}\wedge{\rm
d}z^{B}_{-},\qquad\kappa_{A\tilde{B}}=\kappa(\partial_{z^{A}_{+}},\partial_{z^{B}_{-}}).$
(8)

Indices $A,\tilde{B},...$ are numerical and take values $0,1$ (and again
Einstein summation is used). The distinction between an index ‘$B$’ and an
index ‘$\tilde{B}$’ is only intended to remind that they are associated to the
two different halves of the quadric, and in equations like (8) they are summed
over as usual.

##  3 Deformed quadrics and non-self-dual Kähler metrics

Consider a holomorphic quadric $\mathbb{Q}$ in twistor space, which is
arbitrary except for the assumption that, generically, twistor lines intersect
$\mathbb{Q}$ at two points, so that the quadric is divided into two regions
$\mathbb{A}_{\pm}$ as in (4). Choose holomorphic coordinates $z^{A}_{\pm}$ on
$\mathbb{A}_{\pm}$. We now introduce a “deformed” quadric as

$\displaystyle\mathcal{Q}=\mathcal{A}_{+}\cup\mathcal{A}_{-},$ (9)

where $\mathcal{A}_{+}$ and $\mathcal{A}_{-}$ are the level sets of the four
functions $\dot{z}^{A}_{+}$ and $\dot{z}^{A}_{-}$ defined by

$\displaystyle\dot{z}^{A}_{+}=z^{A}_{+},\qquad\dot{z}^{A}_{-}=z^{A}_{-}+f^{A}(z^{B}_{+},z^{B}_{-})$
(10)

for some functions $f^{A}$, such that ${\rm d}\dot{z}^{0}_{+}\wedge{\rm
d}\dot{z}^{1}_{+}\wedge{\rm d}\dot{z}^{0}_{-}\wedge{\rm d}\dot{z}^{1}_{-}\neq
0$. Although one half of the quadric remains “undeformed”,
$\mathbb{A}_{+}=\mathcal{A}_{+}$, the other half $\mathbb{A}_{-}$ is deformed
to $\mathcal{A}_{-}$ and $\mathcal{Q}$ is, in general, not inside twistor
space (a point in $\mathcal{A}_{-}$ is not in twistor space, since it does not
correspond to an $\alpha$-surface in $\mathbb{CM}$). We also note that in
order to get a non-trivial construction, the functions $f^{A}$ must depend on
both $z^{A}_{+}$ and $z^{A}_{-}$, otherwise (10) would just be a
diffeomorphism on the quadric. Although our construction is inspired by the
non-linear graviton, the sense in which (9) is a deformed quadric does not
seem to be the same as the complex-structure-deformations of twistor theory.

Recalling that the complex structure on space-time induced by the original
quadric is given by (6), we associate the deformed quadric to a new complex
structure:

$\displaystyle\dot{J}={\rm i}(\partial_{\dot{z}^{A}_{+}}\otimes{\rm
d}\dot{z}^{A}_{+}-\partial_{\dot{z}^{A}_{-}}\otimes{\rm d}\dot{z}^{A}_{-}).$
(11)

This is an integrable almost-complex structure on the (complexified) space-
time manifold, but it is not Hermitian: it is not compatible with the
Minkowski metric. In particular, unlike the undeformed quadric, the new
vectors $\partial_{\dot{z}^{A}_{+}}$ are not null (they are linear
combinations of $\partial_{z^{A}_{+}}$ and $\partial_{z^{A}_{-}}$). We then
interpret the deformation (10) of the quadric as a deformation of the
conformal structure on space-time: new conformal structures are introduced by
requiring that their null cones contain $\partial_{\dot{z}^{A}_{\pm}}$.

This requirement alone, however, does not fix a metric. In order to do this,
we must ask additional conditions on the deformations (10). To this end, we
choose to restrict to quadric deformations that preserve the symplectic
structure induced on space-time, $\kappa$. In some sense, we can take
inspiration for this restriction from the non-linear graviton, where the
twistor deformations preserve the fiberwise symplectic structure (which allows
to reconstruct the space-time metric); however, this is not the same since we
are here dealing with a symplectic structure on space-time (not on twistor
space). Regardless, the symplectic-form-preserving condition allows to fix a
metric:

$\displaystyle g(X,Y):=\kappa(X,\dot{J}Y),$ (12)

for all vectors $X,Y$, where the symmetry property of this map follows from
requiring $\kappa$ and $\dot{J}$ to be compatible, which in turn is the same
as requiring $\partial_{\dot{z}^{A}_{\pm}}$ to be null. The metric (12) is
then

$\displaystyle g=2g_{A\tilde{B}}{\rm d}\dot{z}^{A}_{+}\odot{\rm
d}\dot{z}^{B}_{-},$ (13)

where
$g_{A\tilde{B}}=g(\partial_{\dot{z}^{A}_{+}},\partial_{\dot{z}^{B}_{-}})$. As
before, all indices here are numerical, see below eq. (8) for our conventions.
A calculation shows that the deformations (10) preserve the symplectic
structure $\kappa$ if and only if the functions $g_{A\tilde{B}}$ and $f^{A}$
satisfy

$\displaystyle g_{A\tilde{C}}\left(\delta^{C}_{B}+\frac{\partial
f^{C}}{\partial z^{B}_{-}}\right)={}$ $\displaystyle{\rm
i}\kappa_{A\tilde{B}},$ (14a)
$\displaystyle\epsilon^{AC}g_{A\tilde{B}}\frac{\partial f^{B}}{\partial
z^{C}_{+}}={}$ $\displaystyle 0,$ (14b)

where the four functions $\kappa_{A\tilde{B}}$ are defined in (8). The
functions $g_{A\tilde{B}}$ in (13) can then be computed from eq. (14a) (by
inverting the matrix inside the brackets on the left), and the deformation
functions $f^{A}$ are not completely arbitrary but are restricted by the
condition (14b).

In summary: the result of this construction is a new metric (13)-(14) on
space-time that is generically curved, non-(A)SD, and automatically
(complexified) Kähler.

##  4 Black holes

Consider the holomorphic quadric $\mathbb{Q}\subset\mathbb{PT}$ given as the
zero set of the following quadratic function:

$\displaystyle\chi(Z^{\alpha})=Z^{0}Z^{3}-Z^{1}Z^{2}.$ (15)

On twistor lines, this adopts the form (3) with $A=w_{\rm c}$,
$B=\frac{1}{2}(u_{\rm c}-v_{\rm c})$, $C=-\tilde{w}_{\rm c}$ (we are omitting
an irrelevant overall factor of ${\rm i}$ coming from (1)). The roots are then
easily computed to be

$\displaystyle\zeta_{\pm}=\frac{-z_{\rm c}\pm r_{\rm c}}{x_{\rm c}+{\rm
i}y_{\rm c}},\qquad r_{\rm c}:=\sqrt{x_{\rm c}^{2}+y_{\rm c}^{2}+z_{\rm
c}^{2}}.$ (16)

In some sense, we could say that $\zeta_{+}$ and $\zeta_{-}$ are “related” by
a complexified antipodal map: if $\tilde{\zeta}_{\pm}=(-z_{\rm c}\pm r_{\rm
c})/(x_{\rm c}-{\rm i}y_{\rm c})$, then $\zeta_{+}=-1/\tilde{\zeta}_{-}$.
Twistor lines with $r_{\rm c}=0$ intersect $\mathbb{Q}$ only once; the Kähler
structure on $\mathbb{CM}$ is not well-defined at these points. The symplectic
form (7) is

$\displaystyle\kappa=\frac{{\rm i}}{r_{\rm c}^{3}}\left[x_{\rm c}({\rm
d}t_{\rm c}\wedge{\rm d}x_{\rm c}+{\rm i}{\rm d}y_{\rm c}\wedge{\rm d}z_{\rm
c})+y_{\rm c}({\rm d}t_{\rm c}\wedge{\rm d}y_{\rm c}+{\rm i}{\rm d}z_{\rm
c}\wedge{\rm d}x_{\rm c})+z_{\rm c}({\rm d}t_{\rm c}\wedge{\rm d}z_{\rm
c}+{\rm i}{\rm d}x_{\rm c}\wedge{\rm d}y_{\rm c})\right].$ (17)

Recalling (4) and (5), we choose the following quadric coordinates
$z^{0}_{\pm},z^{1}_{\pm}$ on $\mathbb{A}_{\pm}$:

$\displaystyle z^{0}_{\pm}=\omega^{0}_{\pm},\qquad z^{1}_{\pm}={}$
$\displaystyle\tfrac{{\rm i}}{\sqrt{2}}\log(\pm\zeta_{\pm}).$ (18)

We now impose reality conditions: we take the real Lorentzian slice in
$\mathbb{CM}$ defined by

$\displaystyle t_{\rm c}=t,\quad x_{\rm c}=x,\quad y_{\rm c}=y,\quad z_{\rm
c}=z-{\rm i}a$ (19)

where $t,x,y,z$ are all real, and $a$ is a real parameter. The function
$r_{\rm c}$ in (16) is complex: we denote by $r$ its real part, so that (from
the definition of $r_{\rm c}$) we must have $r_{\rm c}=r-{\rm i}az/r$. Let us
introduce a real coordinate system $(r,p,\varphi)$ related to Cartesian
coordinates $(x,y,z)$ by

$\displaystyle x+{\rm i}y=\sqrt{(r^{2}+a^{2})(1-p^{2})}e^{{\rm
i}\varphi},\qquad z=rp$ (20)

(where we assume $p^{2}<1$). The symplectic form (17) and the quadric
coordinates (18) become

$\displaystyle\kappa=\frac{{\rm i}}{(r-{\rm i}ap)^{2}}\left[{\rm d}t\wedge{\rm
d}(r-{\rm i}ap)-{\rm d}\varphi\wedge(a(1-p^{2}){\rm d}r-{\rm
i}(r^{2}+a^{2}){\rm d}p)\right],$ (21) $\displaystyle
z^{0}_{\pm}=\tfrac{1}{\sqrt{2}}[t\pm(r-{\rm i}ap)],\qquad
z^{1}_{\pm}=\tfrac{1}{\sqrt{2}}\left[\varphi\pm\left(-\arctan(a/r)-\tfrac{{\rm
i}}{2}\log\left(\tfrac{1+p}{1-p}\right)\right)\right].$ (22)

After some calculations, we find the components $\kappa_{A\tilde{B}}$ in (8)
to be $\kappa_{0\tilde{0}}={\rm i}r_{\rm c}^{-2}$,
$\kappa_{0\tilde{1}}=0=\kappa_{1\tilde{0}}$, $\kappa_{1\tilde{1}}=-{\rm
i}r_{\rm c}^{-2}(r^{2}+a^{2})(1-p^{2})$, where $r_{\rm c}^{2}=(r-{\rm
i}ap)^{2}$.

Following the prescription (10), we now deform the quadric given by (15) to a
new quadric $\mathcal{Q}=\mathcal{A}_{+}\cup\mathcal{A}_{-}$ according to

$\displaystyle\dot{z}^{A}_{+}=z^{A}_{+},\qquad\dot{z}^{A}_{-}=z^{A}_{-}+R^{A}(r)+P^{A}(p),$
(23)

for arbitrary functions $R^{A}(r),P^{A}(p)$, where $r=r(z^{B}_{+},z^{B}_{-})$
and $p=p(z^{B}_{+},z^{B}_{-})$ are given by inverting the relations (22). A
calculation shows that the symplectic-form-preserving requirement (14b)
reduces to

$\displaystyle(r^{2}+a^{2})\frac{\partial R^{1}}{\partial r}-a\frac{\partial
R^{0}}{\partial r}=0,\qquad a(1-p^{2})\frac{\partial P^{1}}{\partial
p}-\frac{\partial P^{0}}{\partial p}=0$ (24)

so the functions $R^{0},R^{1}$ and $P^{0},P^{1}$ in (23) are not independent
but are related by this condition. The new metric on space-time is given by
(13), (14a), and, as mentioned, it is curved, non-(A)SD, and (complexified)
Kähler. Furthermore, it turns out that this simple prescription already
identifies the Plebański-Demiański class [7, 8]: to see this, we define four
functions ${\rm T}(t,r,p),\Phi(\varphi,r,p),\Delta_{r}(r),\Delta_{p}(p)$ by

$\displaystyle{\rm
T}:=t+\tfrac{1}{\sqrt{2}}(R^{0}(r)+P^{0}(p)),\qquad\Phi:=\varphi+\tfrac{1}{\sqrt{2}}(R^{1}(r)+P^{1}(p)),$
(25)
$\displaystyle\Delta_{r}:=(r^{2}+a^{2})\left(1-\tfrac{1}{\sqrt{2}}\tfrac{\partial
R^{0}}{\partial r}\right)^{-1},\qquad\Delta_{p}:={\rm i}a(1-p^{2})\left({\rm
i}a+\tfrac{1}{\sqrt{2}}\tfrac{\partial P^{0}}{\partial p}\right)^{-1}.$ (26)

After some lengthy calculations, the new metric (13), (14a) is

$\displaystyle g=\frac{1}{r_{\rm c}^{2}}$
$\displaystyle\left[\frac{(\Delta_{r}-a^{2}\Delta_{p})}{\Sigma}{\rm d}{\rm
T}^{2}+\frac{2a[(r^{2}+a^{2})\Delta_{p}-(1-p^{2})\Delta_{r})}{\Sigma}{\rm
d}{\rm T}{\rm d}\Phi\right.$ (27)
$\displaystyle\left.+\frac{[a^{2}(1-p^{2})^{2}\Delta_{r}-(r^{2}+a^{2})^{2}\Delta_{p}]}{\Sigma}{\rm
d}\Phi^{2}-\frac{\Sigma}{\Delta_{r}}{\rm d}r^{2}-\frac{\Sigma}{\Delta_{p}}{\rm
d}p^{2}\right],$

where $\Sigma:=r^{2}+a^{2}p^{2}$. By choosing a specific form for
$\Delta_{r},\Delta_{p}$, this is the Kähler metric associated to the
Plebański-Demiański space-time [13].

We emphasise that the definitions (25)-(26) are introduced only to recover the
familiar form (27): all necessary information about the metric (27) is already
contained in the deformed quadric (23).

As an example, put first $R^{1}=P^{0}=P^{1}=0$ and then $a=0$ (so that
$\Delta_{p}$ reduces to $1-p^{2}$), define $\cos\theta:=p$ and
$f(r):=\Delta_{r}/r^{2}$; then (27) multiplied by $r^{2}$ is the (real,
ordinary) Schwarzschild metric if one sets $\Delta_{r}=r^{2}-2Mr$. Similarly,
the Reissner-Nördstrom metric, and cosmological versions, etc., are obtained
by choosing different functions $\Delta_{r}$. Space-time points with $r=0$,
corresponding to twistor lines intersecting the undeformed quadric
$\mathbb{Q}\subset\mathbb{PT}$ only once, are curvature singularities.

As another example, put $P^{0}=P^{1}=0$. Defining $\cos\theta:=p$, and setting
$\Delta_{r}=r^{2}-2Mr+a^{2}$, the metric (27) multiplied by $r_{\rm c}^{2}$ is
the (real) Kerr metric. The Kerr-Newman metric corresponds to
$\Delta_{r}=r^{2}-2Mr+a^{2}+Q^{2}$, and to obtain the cosmological versions
one must include non-trivial $P^{0},P^{1}$. Twistor lines intersecting
$\mathbb{Q}$ only once are those with $r_{\rm c}=0$, which is the same as
$r=0=\cos\theta$ and correspond to ring singularities.

##  5 Final remarks

The particular quadric (15) used in the derivation of the Plebański-Demiański
space-time can be given some sort of physical interpretation, by writing it as
$Q_{\alpha\beta}Z^{\alpha}Z^{\beta}$ and noticing that $Q_{\alpha\beta}$ is
the angular-momentum twistor corresponding to a static, spin-less particle at
rest in a complex space-time. In fact, the Penrose transform can be used here
to show that the associated twistor functions produce the spin 2 field of
linearized black holes [3]. This idea has been revived in recent interesting
work on scattering amplitudes, see e.g. [14]. Our construction shows, however,
that the exact non-linear solutions are associated to a deformation of the
quadric (which is not inside twistor space), in line with the general twistor
philosophy that curved space-times should correspond to deformed twistor
structures.

The approach in this work has been to take a complexified space-time as a
starting point, and then recover real slices by the imposition of (Lorentzian)
reality conditions. This is why we needed to consider only one quadric (15) to
recover different space-times. If, on the contrary, we assume from the
beginning that $t_{\rm c},x_{\rm c},y_{\rm c},z_{\rm c}$ in (1) are real, then
the quadrics for (say) Schwarzschild and Kerr are different. Also, we have
chosen to work with the form of the Plebański-Demiański space-time given in
[8], as this allows to recover standard black holes in a straightforward
manner. If we wish to work with the original Plebański-Demiański coordinates
[7], one possibility is to start from a twistor quadric different from (15).
The corresponding quadric is $\chi=Z^{0}Z^{1}+cZ^{2}Z^{3}$, as was found by
Haslehurst and Penrose [15].

There are many open questions concerning our construction that we believe
deserve further investigation. Can any (Riemannian or Lorentzian) Kähler
metric be obtained by this procedure? In particular, the Chen-Teo instanton
[16]? Also, the Einstein equations in the non-linear graviton are
automatically encoded in the deformed twistor space; how are field equations
encoded in the deformations considered in this work? It would also be
desirable to obtain a more intrinsic (i.e. not coordinate dependent) approach
to the deformations. Finally, “non-integrable” deformations of the quadric
might be related to black hole perturbation theory, since metric perturbations
constructed from the Teukolsky equations still possess one family (but not
two) of $\alpha$-surfaces [17].

Acknowledgements. I would like to thank S. Aksteiner and L. Andersson for
discussions. The author gratefully acknowledges the support of the Alexander
von Humboldt foundation.

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