# Free-Fermionic Topological Quantum Sensors

Saubhik Sarkar<EMAIL_ADDRESS>Institute for Quantum Science and
Technology and Department of Physics and Astronomy, University of Calgary,
Calgary, Alberta T2N 1N4, Canada  Chiranjib Mukhopadhyay
<EMAIL_ADDRESS>RCQI, Institute of Physics, Slovak Academy of
Sciences, Dúbravská cesta 9, 84511 Bratislava, Slovakia  Institute of
Fundamental and Frontier Sciences, University of Electronic Science and
Technology of China, Chengdu 610051, China  Abhijeet Alase
<EMAIL_ADDRESS>Institute for Quantum Science and Technology and
Department of Physics and Astronomy, University of Calgary, Calgary, Alberta
T2N 1N4, Canada  Abolfazl Bayat<EMAIL_ADDRESS>Institute of
Fundamental and Frontier Sciences, University of Electronic Science and
Technology of China, Chengdu 610051, China

###### Abstract

Second order quantum phase transitions, with well-known features such as long-
range entanglement, symmetry breaking, and gap closing, exhibit quantum
enhancement for sensing at criticality. However, it is unclear which of these
features are responsible for this enhancement. To address this issue, we
investigate phase transitions in free-fermionic topological systems that
exhibit neither symmetry-breaking nor long-range entanglement. We analytically
demonstrate that quantum enhanced sensing is possible using topological edge
states near the phase boundary. Remarkably, such enhancement also endures for
ground states of such models that are accessible in solid state experiments.
We illustrate the results with 1D Su-Schrieffer-Heeger chain and a 2D Chern
insulator which are both experimentally accessible. While neither symmetry-
breaking nor long-range entanglement are essential, gap closing remains as the
major candidate for the ultimate source of quantum enhanced sensing. In
addition, we also provide a fixed and simple measurement strategy that
achieves near-optimal precision for sensing using generic edge states
irrespective of the parameter value. This paves the way for development of
topological quantum sensors which are expected to also be robust against local
perturbations.

_Introduction.—_ The sensitivity of quantum systems to the variation of their
environment makes them excellent sensors Degen  _et al._ (2017). The
uncertainty of measuring an unknown parameter $\lambda$, quantified by
standard deviation $\delta\lambda$ is bounded by Cramér-Rao inequality
$\delta\lambda\geq 1/\sqrt{\mathcal{M}F}$, where $\mathcal{M}$ is the number
of trials and $F$ is the Fisher information Paris (2009). In a classical
setup, the Fisher information scales linearly with the sensor size (known as
standard limit). However, quantum features, such as superposition, may enhance
the resource efficiency of a quantum sensor such that the Fisher information
scales quadratically with system size (known as the Heisenberg limit) Paris
(2009), or even faster (super-Heisenberg limit) Rams  _et al._ (2018); Gong
and Tong (2008); Gu  _et al._ (2008); Greschner  _et al._ (2013); Mishra and
Bayat (2021a); Garbe  _et al._ (2021). There are at least two major approaches
for achieving quantum enhanced sensing: (i) exploiting GHZ-type entangled
states Giovannetti  _et al._ (2004, 2006); Fröwis and Dür (2011); Demkowicz-
Dobrzański  _et al._ (2012); Wang  _et al._ (2018); Kwon  _et al._ (2019) for
estimating the angle of a unitary rotation Rams  _et al._ (2018); and (ii)
utilizing quantum criticality for directly estimating the Hamiltonian
parameters Zanardi and Paunković (2006); Zanardi  _et al._ (2007, 2008);
Invernizzi  _et al._ (2008); Gu (2010); Gammelmark and Mølmer (2011); Chu  _et
al._ (2021); Liu  _et al._ (2021); Montenegro  _et al._ (2021); Skotiniotis
_et al._ (2015). In the former, the interaction between the particles in the
quantum sensor degrades the sensing quality Boixo  _et al._ (2007); De
Pasquale  _et al._ (2013); Skotiniotis  _et al._ (2015); Pang and Brun (2014).
Also, because of extreme vulnerability of GHZ states to decoherence and
particle loss, it is difficult to be scaled up Kołodyński and Demkowicz-
Dobrzański (2013). In the latter, however, the interaction between the
constituents of the quantum sensor is crucial and the system is more robust
against decoherence. Originally the criticality-enhanced quantum sensing has
been introduced for the ground state of many-body systems undergoing a second
order quantum phase transition Zanardi and Paunković (2006); Zanardi  _et al._
(2007, 2008); Invernizzi  _et al._ (2008); Skotiniotis  _et al._ (2015); Gu
(2010); Gammelmark and Mølmer (2011); Liu  _et al._ (2021); Chu  _et al._
(2021); Montenegro  _et al._ (2021). In such symmetry breaking transitions,
the ground state reveals long-range correlations which lead to the scaling of
$F\sim V^{2/D\nu}$, where $V$ is the system size (volume), $D$ is the
dimension, and $\nu$ is the critical exponent with which the correlation
length diverges near the criticality Rams  _et al._ (2018). Recently, quantum
enhanced sensing has also been observed in integrable Floquet systems Mishra
and Bayat (2021a, b) along the line that the Floquet gap vanishes. An
important open question is what feature of a phase transition, e.g. symmetry
breaking, long-range correlations, or vanishing gap, is truly responsible for
obtaining quantum enhanced sensing.

To answer this, one needs to investigate the scaling of Fisher information in
different types of quantum phase transitions, e.g., those not of the symmetry-
breaking kind. Phase transitions in symmetry-protected topological (SPT)
phases of noninteracting fermions Kitaev (2009); Ryu  _et al._ (2010) are
ideal candidates for this investigation. These topological phase transitions
(TPT) are fundamentally different from the symmetry-breaking ones in at least
three aspects Bernevig (2013). First, a fermionic SPT phase transition
manifests in the form of robust edge/surface states protected against
symmetry-preserving local perturbations Alldridge  _et al._ (2020); Alase
(2019). Second, they are not detected by a local order parameter, but rather
by an integer-valued nonlocal quantity called a topological invariant Bernevig
(2013). Third, unlike the symmetry-breaking phase transitions, the fermionic
SPT phases at TPT are short-range entangled Chen  _et al._ (2013). These
differences between second-order quantum phase transitions and fermionic SPT
phase transitions motivate our investigation of quantum enhanced sensitivity
in the latter. In fact, sensing based on non-Hermitian systems Wiersig (2014)
including topological systems Schomerus (2020); Budich and Bergholtz (2020);
Koch and Budich (2022) and TPTs for rotation angle estimation (Pezze  _et al._
, 2017; Zhang  _et al._ , 2018; Lambert and Sørensen, 2020; Yin  _et al._ ,
2019; Chen  _et al._ , 2021; Zhang  _et al._ , 2022; Yang  _et al._ , 2022)
(like GHZ state-based metrology) have been already proposed. Nonetheless, the
sensing capability of free-fermionic TPTs for estimating Hamiltonian
parameters is yet to be explored. Fermionic SPT phases have been realized with
solid-state systems Klitzing  _et al._ (1980); Bernevig  _et al._ (2006) and
simulated platforms Cooper  _et al._ (2019); Ozawa  _et al._ (2019).
Therefore, finding quantum enhanced precision in such systems is a key step
forward for developing topological quantum sensors.

In this Letter, we analytically address the quantum sensing capability of
free-fermionic topological systems. We have two main findings. First, from a
practical perspective, we show that these systems indeed reveal quantum
enhanced sensitivity and thus are legitimate candidates for developing
topological quantum sensors naturally robust against local perturbations.
Second, from a fundamental perspective, we highlight the importance of gap
closing, as opposed to symmetry-breaking or long-range entanglement, for
quantum enhanced sensing.

_Ultimate precision limit.—_ To infer an unknown parameter $\lambda$, encoded
in a quantum state $\rho_{\lambda}$, one has to perform measurement on the
system and then feed the outcomes into an estimator algorithm. For a basic
introduction to single parameter estimation, we refer to Supplemental Material
(SM) Sup . For a given measurement setup, described by a set of projective
operators $\\{\Pi_{n}\\}$, every outcome appears with the probability
$p_{n}(\lambda)=\text{Tr}\left[\rho_{\lambda}\Pi_{n}\right]$. In this case,
all the information is encoded in a classical probability distribution and
thus the Cramér-Rao bound is determined by classical Fisher information (CFI),
defined as $F^{C}{=}\sum_{k}p_{n}(\partial_{\lambda}\log p_{n})^{2}$. One can
maximize the CFI for all possible measurement setups to obtain quantum Fisher
information (QFI) as the ultimate precision bound. The QFI can be computed as
$F^{Q}=\text{Tr}\left[\mathcal{L}_{\lambda}^{2}\rho_{\lambda}\right]$, where
$\mathcal{L}_{\lambda}$ is the symmetric logarithmic derivative (SLD) operator
defined as
$\partial_{\lambda}\rho_{\lambda}=(\rho_{\lambda}\mathcal{L}_{\lambda}+\mathcal{L}_{\lambda}\rho_{\lambda})/2$.
For pure states $\rho_{\lambda}=\ket{\psi_{\lambda}}\bra{\psi_{\lambda}}$, SLD
operator simplifies to
$\mathcal{L}_{\lambda}=2\partial_{\lambda}\rho_{\lambda}$, and
$F^{Q}=4\left(\braket{\partial_{\lambda}\psi_{\lambda}}{\partial_{\lambda}\psi_{\lambda}}-|\braket{\partial_{\lambda}\psi_{\lambda}}{\psi_{\lambda}}|^{2}\right)$
(Paris, 2009). It is worth emphasizing that the optimal measurement setup that
achieves the ultimate precision bound is not unique, although one solution is
always given by the eigenvectors of the SLD operator.

_Free-fermionic SPT model.—_ Free-fermionic SPT phases host energy excitations
localized on the boundary known as edge/surface states. The existence of these
states is guaranteed by the nontrivial topology of the filled band wave
functions Alldridge  _et al._ (2020). These states are studied using tight-
binding models Alase  _et al._ (2016, 2017); Cobanera  _et al._ (2017, 2018)
(see SM Sup  for more details). We first analyze the QFI of the edge states in
1D systems and later generalize our results to higher dimensions. Consider a
1D lattice with sites labeled by $\\{j:j{\in}[L]\\}$, where
$[L]{=}\\{0,1,\dots,L-1\\}$. Suppose there are $d$ internal degrees of freedom
associated to each lattice site. The single-particle Hilbert space
$\mathscr{H}$ is then spanned by orthonormal basis states
$\\{\ket{j,m}:j{\in}[L],m{\in}[d]\\}$, and can be tensor-factorized as
$\mathscr{H}{\cong}\mathscr{H}_{L}{\otimes}\mathscr{H}_{I}$ Alase  _et al._
(2016, 2017). The single-particle Hamiltonian of a number-conserving,
noninteracting fermionic system with uniform coupling and open boundary
conditions (OBC) can be expressed as

$H{=}\sum_{j\in[L]}\ket{j}\bra{j}\otimes
h_{0}(\lambda)+\sum_{j<j^{\prime}\in[L]}\left(\ket{j}\bra{j^{\prime}}\otimes
h_{j^{\prime}-j}(\lambda)+\text{H.c.}\right)\,,$ (1)

where each $h_{j^{\prime}-j}$ is a $d{\times}d$ matrix whose entries are
complex amplitudes of hopping between lattice sites separated by distance
$j^{\prime}{-}j$ possibly accompanied by a change in internal state Alase  _et
al._ (2016, 2017), and these entries depend on the parameter $\lambda$ that is
being estimated. Hamiltonians of the form in Eq. (1) are routinely used for
investigation of edge states in 1D free-fermionic topological systems. Our
analysis can be generalized to number nonconserving systems including Kitaev
chain Kitaev (2001), as Bogoliubov-de Gennes Hamiltonian of such systems has
the same structure as in Eq. (1).

_Edge states in 1D.—_ The zero-energy edge states (topologically protected or
accidental) localized on the $j=0$ edge are described (or well-approximated)
by $\ket{\psi_{\rm edge}}=\ket{\phi_{z}}\ket{u}$, where

$\ket{\phi_{z}}=\sqrt{\frac{1-|z|^{2}}{1-|z|^{2L}}}\sum_{j\in[L]}z^{j}\ket{j},\quad
z\in\mathbb{C},\ |z|<1$ (2)

parametrized by $z$ accounts for the spatial part of exponentially decaying
nature of $\ket{\psi_{\rm edge}}$, and $\ket{u}\in\mathscr{H}_{I}$ is an
internal state vector Alase  _et al._ (2016, 2017); Cobanera  _et al._ (2017).
Both $z$ and $\ket{u}$ depend on $\lambda$ in general.

We now derive the scaling of QFI for $\ket{\psi_{\rm edge}}$, assuming
$\arg(z)$ is independent of $\lambda$, and leave the general case for SM Sup .
Our results are stated using $O$, $\Omega$, and $\Theta$ asymptotic notations,
to denote upper, lower, and tight bounds on the scaling respectively Brassard
and Bratley (1996). The QFI for $\ket{\psi_{\rm edge}}$ with respect to
$\lambda$ can be expressed as $F_{\ket{\psi_{\rm
edge}}}(\lambda){=}F_{\ket{\phi_{z}}}(\lambda)+F_{\ket{u}}(\lambda)$. For
$L{\gg}1$, both $z$ and $\ket{u}$ approach a fixed value that does not depend
on $L$ Alase  _et al._ (2016, 2017). However, the state $\ket{\phi_{z}}$
depends on $L$ due to the normalization. Therefore, the scaling of QFI comes
from the scaling of
$F_{\ket{\phi_{z}}}(\lambda){=}4\left(\braket{\partial_{\lambda}\phi_{z}}{\partial_{\lambda}\phi_{z}}-|\braket{\partial_{\lambda}\phi_{z}}{\phi_{z}}|^{2}\right)$.
For $\arg(z)$ independent of $\lambda$,
$\braket{\partial_{\lambda}\phi_{z}}{\phi_{z}}{=}0$, and simple algebra
reveals

$F_{\ket{\phi_{z}}}(\lambda)=\frac{4(\partial_{\lambda}|z|)^{2}[1+|z|^{4L}-|z|^{2L-2}(2|z|^{2}+L^{2}(1-|z|^{2})^{2})]}{(1-|z|^{2})^{2}(1-|z|^{2L})^{2}}\,.$
(3)

Away from TPT, $|z|{<}1$ yields
$\text{lim}_{L\to\infty}F_{\ket{\phi_{z}}}(\lambda)=4(\partial_{\lambda}z)^{2}(1-z^{2})^{-2}$,
so that $F_{\ket{\psi_{\rm edge}}}(\lambda)\in\Theta(1)$. As edge states are
localized single particle excitations, we do not expect $L$-dependent scaling
away from TPT. In contrast, at TPT, the zero-energy edge states undergo
delocalization, so that $|z|\to 1$ as $\lambda$ approaches the transition
point $\lambda_{\rm c}$ Alase  _et al._ (2016, 2017). Consequently, by
calculating the limit of Eq. (3) as $\lambda\to\lambda_{\rm c}$, we get

$\lim_{\lambda\to\lambda_{\rm
c}}F_{\ket{\phi_{z}}}(\lambda){=}\frac{(\partial_{\lambda}z)^{2}(L^{2}{-}1)}{3}{\implies}F_{\ket{\psi_{\rm
edge}}}(\lambda_{\rm c})\in\Theta(L^{2})\,,$ (4)

independent of the model Hamiltonian. The same result holds for complex $z$,
as we show in the SM Sup . This quadratic scaling of the QFI of edge states at
the phase transition is a remarkable observation showing the power of free-
fermionic topological systems for achieving quantum enhanced sensitivity. This
is in fact the first main result of our work.

_Edge states in higher dimensions.—_ We now investigate the scaling of the QFI
of the edge states of $D$-dimensional systems in which periodic boundary
conditions (PBC) are enforced along $D{-}1$ directions, and OBC along the
remaining direction. For ease of explanation, consider a 2D square lattice
with the orthonormal basis states
$\\{\ket{j_{1},j_{2},m}:j_{1}\in[L_{1}],j_{2}\in[L_{2}],m\in[d]\\}$ Cobanera
_et al._ (2018). A Hamiltonian with PBC along both the spatial directions can
be expressed as $H_{\rm PBC}{=}\oplus_{\bm{k}}H_{\bm{k}}$ with $\bm{k}$ in the
2D Brillouin zone and $H_{\bm{k}}$ the Bloch Hamiltonian. If OBC is enforced
along the first spatial direction, then $\bm{k}$ is no longer a good quantum
number. However, $k_{\parallel}$ (component of $\bm{k}$ along the periodic
direction) remains a good quantum number, and therefore the total Hamiltonian
can be expressed as $H_{\rm OBC}{=}\oplus_{k_{\parallel}}H_{k_{\parallel}}$,
with $H_{k_{\parallel}}$ denoting the Hamiltonian of a virtual 1D wire labeled
by $k_{\parallel}$ Cobanera  _et al._ (2018). Each $H_{k_{\parallel}}$ has a
structure similar to that of the Hamiltonian in Eq. (1). An edge state
$\ket{\psi_{\rm edge}}$ at a fixed $k_{\parallel}$ is well-approximated by
$\ket{\psi_{\rm
edge}}{=}\ket{k_{\parallel}}\ket{\phi_{z}}\ket{u(z,k_{\parallel})}$, where
$\ket{\phi_{z}}$ is given in Eq. (2) with the replacement $L{\to}L_{2}$, and
$\ket{k_{\parallel}}{=}\frac{1}{\sqrt{L_{1}}}\sum_{j_{1}\in[L_{1}]}e^{ik_{\parallel}j_{1}}\ket{j_{1}}$
with $k_{\parallel}{\in}[-\pi,\pi)$. For an edge state at a fixed value of
$k_{\parallel}$, we have $\partial_{\lambda}\ket{k_{\parallel}}{=}0$, and
therefore $F_{\ket{\psi_{\rm
edge}}}(\lambda)=F_{\ket{\phi_{z}}}(\lambda)+\text{constant}\in\Theta(L_{2}^{2})$
at TPT as in the 1D case. Interestingly, for $L_{1}{=}L_{2}{=}L$, we have
$F_{\ket{\psi_{\rm edge}}}(\lambda)\in\Theta(V)$ where $V{=}L^{2}$ is the
total system size (area). In $D$-dimensions, lattice sites are indexed by
$\\{j_{1},\dots,j_{D}\\}$, and PBC are enforced along the first $D{-}1$
directions. Similar analysis as above yields $F_{\ket{\psi_{\rm
edge}}}(\lambda)\in\Theta(V^{2/D})$ at TPT, similar to the behaviour of QFI at
second order phase transitions Rams  _et al._ (2018). This establishes the
scaling of QFI of edge states in any spatial dimension.

_Optimal measurement basis for edge states.—_ While QFI determines the
ultimate precision bound, its saturation in the Cramér-Rao inequality relies
on the choice of an optimal measurement basis. For the case where $\arg(z)$ is
independent of $\lambda$, the position measurement in the basis

$\mathcal{B}=\left\\{\ket{j}\bra{j}\otimes\mathds{1}_{d},\ j\in[L]\right\\}$
(5)

is sufficient to saturate the Cramér-Rao bound for QFI
$F_{\ket{\phi_{z}}}(\lambda)$ for every $\lambda$. We note that for the
generic QFI expression $F_{\ket{\psi_{\rm
edge}}}(\lambda){=}F_{\ket{\phi_{z}}}(\lambda)+F_{\ket{u}}(\lambda)$, only the
first term contributes to scaling. The second term $F_{\ket{u}}$, coming from
the intrasite physics, does not depend on lattice size and approaches a fixed
value for $L\gg 1$. Consequently, the measurement of $\ket{\psi_{\rm edge}}$
performed in this basis yields optimal precision up to a length-independent
constant. The proof, obtained by showing that QFI for $\ket{\phi_{z}}$ equals
the CFI in this basis, is detailed in the SM (Sup, ). Physically, this entails
measuring the location of the particle in the lattice. We emphasize that this
measurement basis, independent of parameter values and obviously site-local,
is in sharp contrast to many proposals of quantum many-body sensors, where
Heisenberg scaling is achievable in theory, but only through highly non-local,
complicated, and parameter-sensitive measurement bases. Even dropping the
assumption of $\arg(z)$ being independent of $\lambda$, we show in the SM
(Sup, ) that this measurement basis still yields quadratic scaling of QFI,
which lends generality to quantum-enhanced sensitivity.

At this point we note that the localization feature of the edge states is not
necessary for quadratic scaling of QFI at TPT. For example, as we later show
numerically, quadratic scaling is observed for the bulk states at the top of
the lower energy band as well as bottom of the upper energy band as we
approach TPT from the trivial phase. However, for such states, the measurement
basis in Eq. (5) may not be optimal.

_Example for 1D.—_ As a concrete example we consider the Su-Schrieffer-Heeger
(SSH) Hamiltonian Su  _et al._ (1979)

$\displaystyle\hat{H}^{\rm
SSH}=-\sum_{j\in[L]}\left(J_{1}\hat{b}_{j}^{{\dagger}}\hat{a}_{j}+J_{2}\hat{a}_{j+1}^{{\dagger}}\hat{b}_{j}+\text{H.c.}\right)\,,$
(6)

where $J_{1}$ and $J_{2}$ are the exchange couplings of the two internal
states (denoted by fermionic operators $\hat{a}_{j}$ and $\hat{b}_{j}$ at site
$j$) at a single site and between adjacent sites, respectively. This
Hamiltonian is of the same form as in Eq. (1), with only nonzero matrices
$h_{0}{=}-J_{1}\sigma_{x}$, and $h_{1}{=}-J_{2}(\sigma_{x}-i\sigma_{y})/2$,
with $\sigma_{x},\sigma_{y}$ being Pauli matrices. This model exhibits TPT at
$J_{1}{=}J_{2}$ protected by sublattice symmetry, and has been realized in
both solid state Su  _et al._ (1979) and optical lattice Atala  _et al._
(2013) experiments. For simplicity, we shall assume that $\hat{b}_{L-1}$ is
isolated from other sites in the SSH chain 111This assumption leads to special
boundary conditions as discussed in Appendix E of Ref. Cobanera  _et al._
(2018). These boundary conditions are different from the open boundary
conditions discussed in Eq. (1), but it does not change the form of edge
states in Eq. (2)., and $\lambda{=}J_{1}/J_{2}$ is a real parameter which has
to be estimated. For $\lambda{<}\lambda_{\rm c}=1$, the normalized edge state
solution is given by Zaimi  _et al._ (2021) $|\psi_{\rm edge}^{\rm
SSH}\rangle{=}\ket{\phi_{z=-\lambda}}\ket{u}$, where $\ket{\phi_{z}}$ is given
by Eq. (2) and $\ket{u}=[1\enspace 0]^{T}$. We obtain the same scaling
relation for QFI as in the general case. To verify this numerically, one can
use a fit function $aL^{b}+c$ to the QFI of $\ket{\psi_{\rm edge}^{\rm SSH}}$
for each value of $\lambda$ and extract the exponent $b$. For the trivial
regime ($\lambda>\lambda_{\rm c}$), the edge state smoothly deforms into the
bulk state at the top of the lower energy band, which we use to calculate the
QFI scaling. As displayed in Fig. 1(a), scaling of QFI changes from quadratic
($b{=}2$) to constant ($b{=}0$), as one moves away from TPT. Note that $z$ is
real and $\ket{u}$ is constant, therefore the measurement described by Eq. (5)
is optimal.

---
Figure 1: Scaling exponent of QFI of edge state (for $\lambda<\lambda_{\rm
c}$) and corresponding bulk state (for $\lambda>\lambda_{\rm c}$) as a
function of $\lambda$ for (a) SSH model, and (b) Chern insulator model.

_Example for 2D.—_ We now illustrate the scaling of QFI for a Chern insulator
on a square lattice – a prototype of topological insulators with broken time-
reversal symmetry Ryu  _et al._ (2010). It has also been experimentally
realized in optical lattices Wu  _et al._ (2016). The spin-orbit coupled (SOC)
Hamiltonian is Sticlet  _et al._ (2012); Zhang  _et al._ (2017)

$\displaystyle\hat{H}^{\rm
Ch}=\displaystyle\sum_{\bm{k}}\begin{bmatrix}\hat{c}^{{\dagger}}_{\bm{k},\uparrow}&\hat{c}^{{\dagger}}_{\bm{k},\downarrow}\end{bmatrix}\enspace
H_{\bm{k}}^{\rm
Ch}\enspace\begin{bmatrix}\hat{c}_{\bm{k},\uparrow}&\hat{c}_{\bm{k},\downarrow}\end{bmatrix}^{T}\,,$
(7)

where $H_{\bm{k}}^{\rm Ch}=\bm{B}\cdot\bm{\sigma}$ is the Bloch Hamlitonian
with
$\bm{B}=\left(2t_{1}\cos{k_{x}},\,2t_{1}\cos{k_{y}},\,m_{z}+2t_{2}(\sin{k_{x}}+\sin{k_{y}})\right)$
and $\bm{\sigma}$ the vector of Pauli matrices. Here $\uparrow,\downarrow$
denote spin-$1/2$ up and down states, and $m_{z},t_{1},t_{2}$ are lattice
parameters. We will consider $\lambda{=}m_{z}/t_{2}$ as the parameter to be
estimated. The eigenvectors form two bands that touch at phase transition at
the Dirac points $(k_{x},k_{y}){=}\pm(\pi/2,\pi/2)$ for nonzero $\lambda$, and
the phase boundaries are given by $\lambda_{\rm c}=\mp 4$ Zhang  _et al._
(2017). We impose PBC along $x$ direction ($k_{\parallel}=k_{x}$), and
decompose the Hamiltonian as $H^{\rm Ch}{=}\oplus_{k_{x}}H_{k_{x}}$, where
$H_{k_{x}}$ describes a virtual 1D wire Hamiltonian of the form in Eq. (1)
with
$h_{0}(k_{x})=2t_{1}\cos{k_{x}}\sigma_{x}+(m_{z}+2t_{2}\sin{k_{x}})\sigma_{z},h_{1}(k_{x})=t_{1}\sigma_{y}-it_{2}\sigma_{z}$.
The QFI of the edge state at $k_{x}=\pi/2$ localized near $j=0$, displayed in
Fig. 1(b), shows quadratic scaling at TPT and constant scaling away from it.
As before, the quadratic scaling is shown to be approached from the trivial
phase as well for the corresponding bulk state.

_QFI of many-body ground state.—_ We now look at the scaling nature for the
fermionic many-body ground states, which are relevant for solid state
experiments. We first derive a formula for the QFI of a general many-body
state $\ket{\Psi}$ of $N$ fermions occupying single-particle states denoted by
$\ket{\psi_{1}},\dots,\ket{\psi_{N}}$. The antisymmetrized wave function for
this state is given by the Slater determinant formula Fetter and Walecka
(2003)
$\ket{\Psi}=(1/\sqrt{N!})\sum_{\sigma\in{S_{N}}}\text{sgn}(\sigma)\ket{\psi_{\sigma_{1}}}\dots\ket{\psi_{\sigma_{N}}}$
where $S_{N}$ is the symmetric group. The QFI of this state, with
$P=\sum_{l=1}^{N}\ket{\psi_{l}}\bra{\psi_{l}}$ as projector on the occupied
states, simplifies to (see SM Sup )

$\displaystyle
F_{\ket{\Psi}}=4\sum_{l=1}^{N}\braket{\partial_{\lambda}\psi_{l}}{\mathds{1}-P}{\partial_{\lambda}\psi_{l}}.$
(8)

We now analytically derive the scaling of QFI under PBC, and later numerically
validate similar results for OBC. Consider the ground state in the
$D$-dimensional case under PBC, with filled lowest band and empty higher
bands. Translational invariance dictates that each single-particle state in
the filled band is of the form
$\ket{\psi_{\bm{k}}}{=}\ket{\bm{k}}\ket{u_{\bm{k}}}$, where $\ket{\bm{k}}$ is
the plane-wave state. Using
$\ket{\partial_{\lambda}\psi_{\bm{k}}}{=}\ket{\bm{k}}\ket{\partial_{\lambda}u_{\bm{k}}}$,
Eq. (8) simplifies to

$\displaystyle F_{\rm GS}^{\rm
PBC}{=}4\sum_{\bm{k}}\left(\braket{\partial_{\lambda}u_{\bm{k}}}{\partial_{\lambda}u_{\bm{k}}}-|\braket{\partial_{\lambda}u_{\bm{k}}}{u_{\bm{k}}}|^{2}\right){=}\sum_{\bm{k}}F_{\ket{u_{\bm{k}}}}\,.$
(9)

_Many-body QFI at TPT.—_ We now show how $\Omega(L^{2})$ scaling of $F_{\rm
GS}^{\rm PBC}$ emerges at TPT in a simplistic model of band-gap inversion in
1D systems Bernevig (2013). Consider a two-band Hamiltonian that can be
approximated as $H_{k}=\alpha k\sigma_{x}+(\lambda-\lambda_{\rm c})\sigma_{z}\
$ near the Dirac point $k=0$, with $\alpha$ a Hamiltonian parameter
independent of $\lambda$ and $L$. The two energy bands touch at $k{=}0$ at TPT
(i.e. $\lambda{=}\lambda_{\rm c}$). Major contribution to QFI is expected from
the states near the Dirac point. In fact, as shown in SM Sup , it is enough to
consider only the two lowest $k$-states to establish a $\Theta(L^{2})$ scaling
for QFI. This, combined with Eq. (9), rules out subquadratic scaling of
$F_{\rm GS}^{\rm PBC}$, so that $F_{\rm GS}^{\rm PBC}\in\Omega(L^{2})$.

To explicitly see this scaling behaviour at TPT we look at our prototypical
examples of 1D SSH chain and 2D Chern insulator mentioned before. In the first
case (see SM Sup )

$\displaystyle F_{\rm PBC}^{\rm SSH}(\lambda_{\rm
c})=\sum_{\kappa=1}^{L-1}\frac{\cot^{2}(\pi\kappa/L)}{4}=\frac{L^{2}-3L+2}{12}\,,$
(10)

which clearly shows the $\Theta(L^{2})$ scaling for large $L$. Moreover, the
Fock basis is an optimal measurement basis, as the ground state of SSH
Hamiltonian has real coefficients in that basis Zaimi  _et al._ (2021). Such a
measurement can be performed by measuring the number operator
$\hat{c}^{\dagger}_{j,m}\hat{c}_{j,m}$ for each fermionic mode.

For the Chern insulator on a $L{\times}L$ lattice, QFI at TPT is Sup

$\displaystyle F_{\rm PBC}^{\text{Ch}}(\lambda_{\rm
c})=\displaystyle\sum_{\bm{k}\neq(\pi/2,\pi/2)}\frac{B_{x}^{2}+B_{y}^{2}}{4(B_{x}^{2}+B_{y}^{2}+B_{z}^{2})^{2}}\,.$
(11)

As we show later, this sum also shows $\Omega(L^{2})$ dependence.

_Many-body QFI away from TPT.—_ To see the scaling of $F_{\rm GS}^{\rm PBC}$
away from TPT, we prove that $F_{\ket{u_{k}}}$ is bounded by a constant
independent of $N$. Therefore, $F_{\rm GS}^{\rm PBC}\in O(N)$, by Eq. (9). We
first observe that $F_{\ket{u_{k}}}\leq
4\braket{\partial_{\lambda}u_{k}}{\partial_{\lambda}u_{k}}$. First-order
perturbation theory yields
$\braket{\partial_{\lambda}u_{k}}{\partial_{\lambda}u_{k}}=|\braket{v_{k}}{\partial_{\lambda}H}{u_{k}}|^{2}/(\epsilon_{1,k}-\epsilon_{0,k})^{2}$,
where $\ket{v_{k}}$ is the higher band wavefunction, and $\epsilon_{0,k}$
($\epsilon_{1,k}$) is the lower (higher) band energy eigenvalues. Now we can
bound $F_{\ket{u_{k}}}$ by
$4\braket{\partial_{\lambda}u_{k}}{\partial_{\lambda}u_{k}}\leq
4\lVert\partial_{\lambda}H_{k}\rVert^{2}/\Delta E^{2}$, where $\Delta E$ is
the band gap, and $\lVert\bullet\rVert$ denotes the operator norm.
Furthermore, $\lVert
H_{k}\rVert\leq\text{sup}\lVert\partial_{\lambda}H_{k}\rVert=\lVert\partial_{\lambda}H\rVert$,
hence
$\braket{\partial_{\lambda}u_{k}}{\partial_{\lambda}u_{k}}\leq\lVert\partial_{\lambda}H\rVert^{2}/\Delta
E^{2}$. This proves that QFI of $\ket{\Psi^{\rm PBC}_{\rm GS}}$ scales at most
linearly with the system size ($O(N)=O(V)$) away from TPT. This is in stark
contrast with the constant scaling of QFI for the edge states.

---
Figure 2: Scaling exponent of QFI of many-body ground state as a function of
$\lambda$ subject to PBC (blue) and OBC (red) for (a) SSH model, and (b) Chern
insulator model.

Linear scaling away from TPT can be explicitly proved for SSH model in the
continuum limit $L{\rightarrow}\infty$ (see SM (Sup, )), as

$\lim_{L\to\infty}\frac{F_{\rm
PBC}^{\text{SSH}}(\lambda)}{L}=\begin{cases}1/2(1-\lambda^{2})&\quad\text{if
}\lambda<1\\\ 1/2(\lambda^{4}-\lambda^{2})&\quad\text{if }\lambda>1.\\\
\end{cases}$ (12)

We further provide numerical confirmations by repeating the fitting procedure
as before. The scaling exponents versus $\lambda$ are shown in Figs. 2(a)-(b)
for the SSH and Chern insulator model respectively. Expectedly, the
$\Omega(L^{2})$ scaling at TPT and $O(L^{D})$ scaling far enough away in the
topological phase are independent of the boundary conditions. We observe
qualitatively similar scaling behaviour in the trivial phase as well. For the
Chern insulator, true OBC are numerically intractable beyond small system
sizes, hence we use strip geometry, which leads to the small discrepancies
with the PBC results.

_Experimental realization.—_ All the ingredients for our proposals are already
present in cold atom experiments in optical lattices. As edge states are
single-particle states they have been observed with both fermions Mancini  _et
al._ (2015) and bosons Stuhl  _et al._ (2015) in quantum Hall systems on
optical lattices using standard imaging techniques for synthetic dimensions by
populating the edge states without populating the bulk. For SSH chain,
proposals for edge state preparation are also in place Barnett (2013);
Krivosenko  _et al._ (2018). Position basis for optimal measurement can be
accessed using quantum gas microscopy Simon  _et al._ (2011). To access the
fermionic many-body state filling up the entire lower band one can bank on the
successful experiments on 1D SOC lattice systems Cheuk  _et al._ (2012); Livi
_et al._ (2016); Kolkowitz  _et al._ (2017). For 2D cases, the fermionic
lattice Hamiltonians are yet to be realized but SOC has been observed in
trapped gases Huang  _et al._ (2016); Meng  _et al._ (2016).

_Conclusion.—_ Through analytical investigation, we show that one can achieve
precision beyond the standard limit at the transition point of free-fermionic
topological models. This paves the way for development of topological quantum
sensors, which are expected to be robust against local perturbations. Our
edge-state based schemes allow achieving Heisenberg-limited sensing via a
simple position measurement, thus avoiding the necessity of complicated highly
entangled optimal measurements that hitherto seemed necessary to build quantum
many-body sensors. From a fundamental point of view, our analysis indicates
that gap closing, rather than long-range entanglement and spontaneous
symmetry-breaking, is essential for obtaining quantum enhanced precision. This
observation is consistent with recent discovery of quantum enhanced
sensitivity at Floquet gap closing Mishra and Bayat (2021a, b) in periodically
driven systems.

###### Acknowledgements.

A. B. acknowledges support from the National Key R&D Program of China (Grant
No. 2018YFA0306703), National Science Foundation of China (Grants No.
12050410253 and No. 92065115) and the Ministry of Science and Technology of
China (Grant No. QNJ2021167001L). S. S. acknowledges support by Alberta Major
Innovation Fund. C. M. acknowledges Slovak Academy of Sciences for funding
from OPTIQUTE APVV-18-0518 and DESCOM VEGA-2/0183/21 projects, and the Stefan
Schwarz Support Fund. A. A. acknowledges support by Killam Trusts
(Postdoctoral Fellowship).

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##  I Supplemental Material: Free-Fermionic Topological Quantum Sensors

Saubhik Sarkar,1 Chiranjib Mukhopadhyay,2,3 Abhijeet Alase,1 and Abolfazl
Bayat3

1Institute for Quantum Science and Technology and Department of Physics and
Astronomy,

University of Calgary, Calgary, Alberta T2N 1N4, Canada

2RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9,
84511 Bratislava, Slovakia

3Institute of Fundamental and Frontier Sciences, University of Electronic
Science and Technology of China, Chengdu 610051, China

###  I.1 Quantum single parameter estimation

Here we give a brief overview of sensing of a single parameter $\lambda$
encoded in a quantum system at state $\rho_{\lambda}$ in terms of the QFI of
the system. To do so, let us first consider a statistical system with
(classical) probability distribution $p(x|\lambda)$ over some random variable
$x$ with outcomes $\\{x_{i}\\}$, when the actual parameter value is $\lambda$.
If one chooses an estimator $\hat{\xi}(x)$ for the parameter $\lambda$ such
that the estimator is unbiased, then this means

$\displaystyle E[\hat{\xi}(x)]=\lambda$
$\displaystyle\Rightarrow\sum_{i}\left(\hat{\xi}(x_{i})-\lambda\right)p(x_{i}|\lambda)=0$
$\displaystyle\Rightarrow\partial_{\lambda}\left[\sum_{i}\left(\hat{\xi}(x_{i})-\lambda\right)p(x_{i}|\lambda)\right]=0$
$\displaystyle\Rightarrow\sum_{i}\left(\hat{\xi}(x_{i})-\lambda\right)\frac{\partial
p(x_{i}|\lambda)}{\partial\lambda}-\sum_{i}p(x_{i}|\lambda)=0$
$\displaystyle\Rightarrow\sum_{i}\left(\hat{\xi}(x_{i})-\lambda\right)\frac{\partial
p(x_{i}|\lambda)}{\partial\lambda}=1$
$\displaystyle\Rightarrow\sum_{i}\left(\hat{\xi}(x_{i})-\lambda\right)p(x_{i}|\lambda)\frac{\partial\log(p(x_{i}|\lambda)}{\partial\lambda}=1$
(S1)

Now, we make use of the Cauchy-Schwarz inequality
$\sum_{i}a_{i}^{2}\sum_{i}b_{i}^{2}\geq(\sum_{i}a_{i}b_{i})^{2}$, where we put
$a_{i}=\left(\hat{\xi}(x_{i})-\lambda\right)\sqrt{p(x_{i}|\lambda)}$ and
$b_{i}=\sqrt{p(x_{i}|\lambda)}\partial\log(p(x_{i}|\lambda)/\partial\lambda$.
From this, we immediately obtain the following lower bound for
$\text{Var}[\hat{\xi}(x)]=\sum_{i}\left(\hat{\xi}(x_{i})-\lambda\right)^{2}p(x_{i}|\lambda)$

$\text{Var}[\hat{\xi}(x)]\geq\frac{1}{\sum_{i}p(x_{i}|\lambda)\left(\log
p(x_{i}|\lambda)\right)^{2}}=\frac{1}{F^{C}(\lambda)},$ (S2)

where $F^{C}$ is the (classical) Fisher information of the system. This
inequality is known in statistics literature as the Cramer-Rao inequality. For
single parameters, it is easy to note that this inequality is saturated
whenever $a_{i}=\left(\hat{\xi}(x_{i})-\lambda\right)\sqrt{p(x_{i}|\lambda)}$
and
$b_{i}=\sqrt{p(x_{i}|\lambda)}\partial\log(p(x_{i}|\lambda)/\partial\lambda$
only differ by a multiplicative constant. In the context of the quantum state
$\rho_{\lambda}$ dependent on the parameter $\lambda$, each measurement
$\Pi=\\{\Pi_{i}\\}$ performed on the system will lead to the situation above
with corresponding probabilities $p_{i}=\text{Tr}[\Pi_{i}\rho]$ and CFI
$F^{C}$ depending on the set of probabilities, and thus, ultimately on the
measurement $\Pi$. QFI $F^{Q}$ is simply defined as the supremal CFI $F^{C}$
over all such measurement bases

$F^{Q}(\lambda)=\sup_{\Pi}F^{C}(\lambda)$ (S3)

It is immediately apparent that finding this optimal measurement allows one to
put tighter lower bounds to the variance of the estimator. This is generally
the main challenge for quantum metrological problems. A method of obtaining
the measurement is via the so-called Symmetric Logarithmic Derivative (SLD)
operator $L$ defined as the solution to the matrix equation
$\partial_{\lambda}\rho=\left(\rho L+L\rho\right)/2$. It can be proved that
$F^{Q}=\text{Tr}[\rho L^{2}]$, thus bypassing the brute-force optimization of
the measurement basis. Moreover, an optimal measurement basis can now be shown
as just the eigenbasis of the SLD operator $L$. The eigenvectors of the SLD
operator therefore always provide an optimal measurement setup that achieves
the ultimate precision bound, although it may not be the only basis to achieve
that.

###  I.2 QFI of edge states with unconstrained $z$

In this work, we consider SPT systems, which possess topological features as
long as the system Hamiltonian preserves certain symmetries, and these
symmetries are said to preserve the topological phase. SPT systems are short-
range entangled and do not require presence of interactions. Over the last two
decades, many free-fermionic SPT systems have been fabricated on solid state
platform, and verified using experimental techniques such as conductance
measurements and angle-resolved photo-emission spectroscopy (ARPES). Other
experimental platforms including ultracold atoms, trapped ions, and photonics
have also achieved great success in simulating SPT phases. In addition to
being experimentally accessible, free-fermionic SPT phases are also
theoretically tractable due to absence of interactions. In fact, an exhaustive
classification of free-fermionic and -bosonic SPT systems in all spatial
dimensions and satisfying various symmetry combinations is well understood.

One of the most important feature of free-fermionic SPT systems is the
presence of protected energy eigenstates localized on their edges/surfaces.
These states play a dominant role in bestowing exotic topological properties,
such as the quantized transverse conductance in integer quantum Hall states.
Such localized energy states are theoretically investigated using tight-
binding models, such as the SSH Hamiltonian and the Chern insulator
Hamiltonian considered in this paper. These localized states persist even in
the presence of weak disorder satisfying the symmetries protecting the
topological phase, and this phenomenon is dubbed as “bulk-boundary
correspondence”. In our work, we propose a sensing procedure that utilizes the
localization properties of these edge states. In the topological phase and
away from the phase boundary, the edge states are exponentially localized near
the boundary. At the topological phase transition, the localization length of
the edge states diverges, thus transforming the edge states into extended
energy states. We show in the present paper that this transition in the
localization length can be used for Heisenberg-limited sensing.

In the main text, the edge state wavefunction is expressed in Eq. (2), in
terms of localization parameter $z$. We have shown the scaling behaviour for
the case where $\arg(z)$ is independent of the parameter to be estimated,
$\lambda$. Now we derive the scaling of QFI for general case where this
constraint is removed. For $z=re^{i\theta}$, we can express
$\ket{\partial_{\lambda}\phi_{z}}=(\partial_{\lambda}r)\ket{\partial_{r}\phi_{z}}+(\partial_{\lambda}\theta)\ket{\partial_{\theta}\phi_{z}}$.
Then the QFI of $\ket{\phi_{z}}$ can be expressed as

$F_{\ket{\phi_{z}}}(\lambda)=\begin{bmatrix}(\partial_{\lambda}r)&(\partial_{\lambda}\theta)\end{bmatrix}\begin{bmatrix}F_{\ket{\phi_{z}}}(r,r)&F_{\ket{\phi_{z}}}(r,\theta)\\\
F_{\ket{\phi_{z}}}(\theta,r)&F_{\ket{\phi_{z}}}(\theta,\theta)\end{bmatrix}\begin{bmatrix}(\partial_{\lambda}r)\\\
(\partial_{\lambda}\theta)\end{bmatrix}\,,$ (S4)

where

$\displaystyle F_{\ket{\phi_{z}}}(r,r)$ $\displaystyle=$ $\displaystyle
4\Big{(}\braket{\partial_{r}\phi_{z}}{\partial_{r}\phi_{z}}-\braket{\partial_{r}\phi_{z}}{\phi_{z}}\braket{\phi_{z}}{\partial_{r}\phi_{z}}\Big{)},$
$\displaystyle F_{\ket{\phi_{z}}}(r,\theta)$ $\displaystyle=$ $\displaystyle
4\Big{(}\braket{\partial_{r}\phi_{z}}{\partial_{\theta}\phi_{z}}-\braket{\partial_{r}\phi_{z}}{\phi_{z}}\braket{\phi_{z}}{\partial_{\theta}\phi_{z}}\Big{)},$
$\displaystyle F_{\ket{\phi_{z}}}(\theta,r)$ $\displaystyle=$ $\displaystyle
4\Big{(}\braket{\partial_{\theta}\phi_{z}}{\partial_{r}\phi_{z}}-\braket{\partial_{\theta}\phi_{z}}{\phi_{z}}\braket{\phi_{z}}{\partial_{r}\phi_{z}}\Big{)},$
$\displaystyle F_{\ket{\phi_{z}}}(\theta,\theta)$ $\displaystyle=$
$\displaystyle
4\Big{(}\braket{\partial_{\theta}\phi_{z}}{\partial_{\theta}\phi_{z}}-\braket{\partial_{\theta}\phi_{z}}{\phi_{z}}\braket{\phi_{z}}{\partial_{\theta}\phi_{z}}\Big{)}\,.$
(S5)

We now use an elegant trick to calculate all quantities on the right-hand side
of Eq. (I.2). First define $\ket{\tilde{\phi_{z}}}=\sum_{j\in[L]}z^{j}\ket{j}$
so that
$\ket{\phi_{z}}=\ket{\tilde{\phi_{z}}}/\big{\|}\ket{\tilde{\phi_{z}}}\big{\|}$.
Some straightforward algebra reveals

$F_{\ket{\phi_{z}}}(r,r)=4\frac{\braket{\partial_{r}\tilde{\phi_{z}}}{\partial_{r}\tilde{\phi_{z}}}}{\big{\|}\ket{\tilde{\phi_{z}}}\big{\|}^{2}}-4\frac{\braket{\partial_{r}\tilde{\phi_{z}}}{\tilde{\phi_{z}}}\braket{\tilde{\phi_{z}}}{\partial_{r}\tilde{\phi_{z}}}}{\big{\|}\ket{\tilde{\phi_{z}}}\big{\|}^{4}}\,,$
(S6)

and similar identities hold for
$F_{\ket{\phi_{z}}}(r,\theta),F_{\ket{\phi_{z}}}(\theta,r)$, and
$F_{\ket{\phi_{z}}}(\theta,\theta)$. Now observe that

$\ket{\partial_{r}{\tilde{\phi_{z}}}}=\sum_{j\in[L]}jr^{j-1}e^{i\theta
j}\ket{j}=\frac{-i}{r}\sum_{j\in[L]}ijr^{j}e^{i\theta
j}\ket{j}=\frac{-i}{r}\ket{\partial_{\theta}{\tilde{\phi_{z}}}}\,.$ (S7)

Therefore, we get

$\displaystyle F_{\ket{\phi_{z}}}(\lambda)$
$\displaystyle=F_{\ket{\phi_{z}}}(r,r)\begin{bmatrix}(\partial_{\lambda}r)&(\partial_{\lambda}\theta)\end{bmatrix}\begin{bmatrix}1&-i/r\\\
i/r&1/r^{2}\end{bmatrix}\begin{bmatrix}(\partial_{\lambda}r)\\\
(\partial_{\lambda}\theta)\end{bmatrix}$
$\displaystyle=F_{\ket{\phi_{z}}}(r,r)[(\partial_{\lambda}r)^{2}+(\partial_{\lambda}\theta)^{2}].$
(S8)

Note that $F_{\ket{\phi_{z}}}(r,r)=F_{\ket{\phi_{z}}}(r)$, and the latter was
calculated in the main text to yield

$F_{\ket{\phi_{z}}}(r)=\frac{4[1+r^{4L}-r^{2L-2}(2r^{2}+L^{2}(1-r^{2})^{2})]}{(1-r^{2})^{2}(1-r^{2L})^{2}}\,.$
(S9)

Away from TPT ($\lambda\neq\lambda_{\rm c}$), we have $r<1$ and therefore
$F_{\ket{\phi_{z}}}(r)\in\Theta(1)$ so that
$F_{\ket{\phi_{z}}}(\lambda)\in\Theta(1)$ as well. As TPT is approached,
($\lambda\to\lambda_{\rm c}$), we have $r\to 1$. In this limit,

$\lim_{r\to 1}F_{\ket{\phi_{z}}}(r)=\frac{L^{2}-1}{3}\,.$ (S10)

Finally substituting this value in Eq. (I.2), we obtain

$\lim_{\lambda\to\lambda_{\rm
c}}F_{\ket{\phi_{z}}}(\lambda)=\frac{(L^{2}-1)[(\partial_{\lambda}r)^{2}+(\partial_{\lambda}\theta)^{2}]}{3}\implies
F_{\ket{\phi_{z}}}(\lambda_{\rm c})\in\Theta(L^{2})\,.$ (S11)

Therefore, it is proven that even for the most general dependence of $z$ on
$\lambda$, QFI of the edge state at TPT shows $\Theta(L^{2})$ scaling.

###  I.3 Optimal measurement for the general edge state

In this section, we show that the projective measurement described by
projectors $\\{\ket{j}\bra{j}\otimes\mathds{1},\ j\in[L]\\}$ is optimal up to
a length-independent prefactor for the general case where $\arg(z)$ depends on
$\lambda$.

We first prove that, if a quantum state $\ket{\psi(\lambda)}$ is expressed in
a basis $\ket{e_{j}}$ has all real coefficients, then QFI
$F^{Q}_{\ket{\psi}}(\lambda)=F^{C}_{\ket{\psi}}(\lambda)$, with
$F^{C}_{\ket{\psi}}(\lambda)$ denoting the CFI with of the probability
distribution $\\{p_{j}=|\braket{\psi}{e_{j}}|^{2}\\}$. To prove this
statement, we express $\ket{\psi}$ as
$\ket{\psi}=\sum_{j}\sqrt{p_{j}}\ket{e_{j}}$, so that
$\ket{\partial_{\lambda}\psi}=\sum_{j}(\partial_{\lambda}p_{j}/2\sqrt{p_{j}})\ket{j}$.
Note that $\braket{\psi}{\partial_{\lambda}\psi}=0$, so that
$F^{Q}_{\ket{\psi}}(\lambda)=4\braket{\partial_{\lambda}\psi}{\partial_{\lambda}\psi}$.
Now

$\displaystyle
F^{Q}_{\ket{\psi}}(\lambda)=4\braket{\partial_{\lambda}\psi}{\partial_{\lambda}\psi}=4\sum_{j}\frac{(\partial_{\lambda}p_{j})^{2}}{4p_{j}}=\sum_{j}p_{j}\left(\frac{\partial_{\lambda}p_{j}}{p_{j}}\right)^{2}=\sum_{j}p_{j}(\partial_{\lambda}\ln
p_{j})^{2}=F^{C}_{\ket{\psi}}(\lambda).$ (S12)

Next, observe that this result holds also under a milder assumption on
$\ket{\psi}$, namely that $\braket{\psi}{e_{j}}$ is complex but
$\arg(\braket{\psi}{e_{j}})$ is independent of $\lambda$. Using this result
for $\ket{\psi}=\ket{\phi_{z}}$ and $\\{\ket{e_{j}}=\ket{j}\\}$, we get QFI
$F_{\ket{\phi_{z}}}(r)=F^{C}_{\ket{\phi_{z}}}(r)$. Then by Eq. (I.2), we get
for the general case, where $\arg(z)$ also depends on $\lambda$,

$F_{\ket{\phi_{z}}}(\lambda)=F^{C}_{\ket{\phi_{z}}}(\lambda)\left(1+\frac{(\partial_{\lambda}\theta)^{2}}{r^{2}(\partial_{\lambda}r)^{2}}\right).$
(S13)

Finally we have QFI for the edge state $F_{\ket{\psi_{\rm
edge}}}(\lambda)=F_{\ket{\phi_{z}}}(\lambda)+F_{\ket{u}}(\lambda)$ and
$F^{C}_{\ket{\psi_{\rm edge}}}(\lambda)=F^{C}_{\ket{\phi_{z}}}(\lambda)$,
which leads to

$F_{\ket{\psi_{\rm edge}}}(\lambda)=F^{C}_{\ket{\psi_{\rm
edge}}}(\lambda)\left(1+\frac{(\partial_{\lambda}\theta)^{2}}{r^{2}(\partial_{\lambda}r)^{2}}\right)+F_{\ket{u}}(\lambda).$
(S14)

Observe that
$\left(1+\frac{(\partial_{\lambda}\theta)^{2}}{r^{2}(\partial_{\lambda}r)^{2}}\right)$
and $F_{\ket{u}}(\lambda)$ do not depend on $L$. Hence, for every value of the
parameter $\lambda$, the position measurement is optimal up to a constant
prefactor and an additive constant independent of $L$.

###  I.4 QFI of many-body Slater determinant states

Starting with the expression for the Slter dterminant state $\ket{\Psi}$ in
the main text we now calculate the QFI using the standard formula

$F_{\ket{\Psi}}=4\left(\braket{\partial_{\lambda}\Psi}{\partial_{\lambda}\Psi}-|\braket{\partial_{\lambda}\Psi}{\Psi}|^{2}\right)\,.$
(S15)

To calculate the first term, we proceed as

$\displaystyle\braket{\partial_{\lambda}\Psi}{\partial_{\lambda}\Psi}$
$\displaystyle=\frac{1}{N!}\sum_{\sigma,\tau\in{S_{N}}}\text{sgn}(\sigma\tau)\sum_{l,l^{\prime}=1}^{N}\Big{(}\bra{\psi_{\tau_{1}}}\dots\bra{\partial_{\lambda}\psi_{\tau_{l}}}\dots\bra{\psi_{\tau_{N}}}\Big{)}\enspace\Big{(}\ket{\psi_{\sigma_{1}}}\dots\bra{\partial_{\lambda}\psi_{\sigma_{l^{\prime}}}}\dots\bra{\psi_{\sigma_{N}}}\Big{)}\,.$
(S16)

Note that if $l=l^{\prime}$, then the non-zero terms correspond to
$\sigma=\tau$. On the other hand, if $l\neq l^{\prime}$, then non-zero terms
correspond to $\sigma=\tau$ and $\sigma=\tau(ll^{\prime})$, where
$(ll^{\prime})$ is the permutation that exchanges the indices $l$ and
$l^{\prime}$. In the latter case, we have $\text{sgn}(\sigma\tau)=-1$. These
observations lead to

$\displaystyle\braket{\partial_{\lambda}\Psi}{\partial_{\lambda}\Psi}$
$\displaystyle=\frac{1}{N!}\sum_{\sigma\in{S_{N}}}\left(\sum_{l=1}^{N}\braket{\partial_{\lambda}\psi_{\sigma_{l}}}{\partial_{\lambda}\psi_{\sigma_{l}}}+\sum_{l\neq
l^{\prime}}\braket{\partial_{\lambda}\psi_{\sigma_{l}}}{\psi_{\sigma_{l}}}\braket{\psi_{\sigma_{l^{\prime}}}}{\partial_{\lambda}\psi_{\sigma_{l^{\prime}}}}-\sum_{l\neq
l^{\prime}}\braket{\partial_{\lambda}\psi_{\sigma_{l}}}{\psi_{\sigma_{l^{\prime}}}}\braket{\psi_{\sigma_{l^{\prime}}}}{\partial_{\lambda}\psi_{\sigma_{l}}}\right)$
$\displaystyle=\sum_{l=1}^{N}\braket{\partial_{\lambda}\psi_{l}}{\partial_{\lambda}\psi_{l}}-\sum_{l\neq
l^{\prime}}\braket{\partial_{\lambda}\psi_{l}}{\psi_{l}}\braket{\partial_{\lambda}\psi_{l^{\prime}}}{\psi_{l^{\prime}}}-\sum_{l\neq
l^{\prime}}\braket{\partial_{\lambda}\psi_{l}}{\psi_{l^{\prime}}}\braket{\psi_{l^{\prime}}}{\partial_{\lambda}\psi_{l}}\,.$
(S17)

We similarly calculate the second term on the right-hand side of Eq. (S15),
which yields

$\displaystyle\braket{\partial_{\lambda}\Psi}{\Psi}^{2}=\left(\sum_{l=1}^{N}\braket{\partial_{\lambda}\psi_{l}}{\psi_{l}}\right)^{2}=\sum_{l\neq
l^{\prime}}\braket{\partial_{\lambda}\psi_{l}}{\psi_{l}}\braket{\partial_{\lambda}\psi_{l^{\prime}}}{\psi_{l^{\prime}}}+\sum_{l}\braket{\partial_{\lambda}\psi_{l}}{\psi_{l}}^{2}\,.$
(S18)

Substituting Eqs. (I.4) and (S18) in Eq. (S15) yields

$\displaystyle F_{\ket{\Psi}}$
$\displaystyle=4\left(\sum_{l=1}^{N}\braket{\partial_{\lambda}\psi_{l}}{\partial_{\lambda}\psi_{l}}-\sum_{l\neq
l^{\prime}}\braket{\partial_{\lambda}\psi_{l}}{\psi_{{l^{\prime}}}}\braket{\psi_{l^{\prime}}}{\partial_{\lambda}\psi_{l}}+\sum_{l}\braket{\partial_{\lambda}\psi_{l}}{\psi_{l}}^{2}\right)$
$\displaystyle=4\sum_{l=1}^{N}\braket{\partial_{\lambda}\psi_{l}}{\mathds{1}-P}{\partial_{\lambda}\psi_{l}}\,,$
(S19)

where $P$ is the projector on the occupied states, i.e.
$P=\sum_{l=1}^{N}\ket{\psi_{l}}\bra{\psi_{l}}$. Now observe that
$(\mathds{1}-P)\ket{\psi_{l}}=0$ for all $l$, hence we can further express

$\displaystyle F_{\ket{\Psi}}$
$\displaystyle=4\text{Tr}\left[(\mathds{1}-P)\left(\sum_{l}\ket{\partial_{\lambda}\psi_{l}}\bra{\partial_{\lambda}\psi_{l}}\right)\right]$
$\displaystyle=2\text{Tr}\left[(\mathds{1}-P)\partial^{2}P\right]\,,$ (S20)

which shows that the many-body QFI can be fully expressed only in terms of the
projection operator.

###  I.5 Many-body QFI scaling at TPT (PBC)

Here we consider the Bloch Hamiltonian from the main text that model the band-
inversion in 1D systems, namely, $H_{k}=\alpha
k\sigma_{x}+(\lambda-\lambda_{\rm c})\sigma_{z}\ $ near the Dirac point $k=0$.
We calculate here the QFI for the two lowest $k$-states which are $k=0$ and
$k=2\pi/L$. At Dirac point, we have $H_{k=0}{=}(\lambda-\lambda_{\rm
c})\sigma_{z}$. For $\lambda\geq\lambda_{\rm c}$, we have,
$\ket{u_{k=0}}=[0\enspace 1]^{T}$, and $\ket{\partial_{\lambda}u_{k=0}}=0$,
resulting in, $F_{\ket{u_{k=0}}}=0$. We next look at the QFI of
$\ket{u_{1}}=\ket{u_{k=2\pi/L}}$ corresponding to $k=2\pi/L$ which is the
closest point to the Dirac point in the Brillouin zone. Here
$H_{k}=\alpha(2\pi/L)\sigma_{x}+(\lambda-\lambda_{\rm c})\sigma_{z}$, so that

$\ket{u_{1}}=\begin{bmatrix}\cos(\gamma/2)\\\
\sin(\gamma/2)\end{bmatrix},\enspace\text{and}\enspace\ket{\partial_{\lambda}u_{1}}=\frac{\partial_{\lambda}\gamma}{2}\begin{bmatrix}-\sin(\gamma/2)\\\
\cos(\gamma/2)\end{bmatrix}\,,$ (S21)

with $\tan\gamma=\alpha/(\lambda-\lambda_{\rm c})$. We now obtain
$F_{\ket{u_{1}}}=(\partial_{\lambda}\gamma)^{2}$. By differentiating the
expression for $\tan\gamma$, we get

$\partial_{\lambda}\gamma=\frac{-\alpha/L}{(\lambda-\lambda_{\rm
c})^{2}+\alpha^{2}/L^{2}}\implies\lim_{\lambda\to\lambda_{\rm
c}}\partial_{\lambda}\gamma=-\frac{L}{\alpha}.$ (S22)

Therefore, at TPT, $F_{\ket{u_{1}}}=L^{2}/\alpha^{2}\in\Theta(L^{2})$.

###  I.6 Many-body QFI scaling at TPT for SSH Hamiltonian (PBC)

The SSH Hamiltonian in Eq. (6) in the main text can rewritten in the momentum
space with the following form of the Bloch Hamiltonian corresponding to
momentum $k=2\pi\kappa/L$

$H_{k}^{\rm SSH}=-\begin{bmatrix}0&J_{1}+J_{2}e^{-i2\pi\kappa/L}\\\
J_{1}+J_{2}e^{i2\pi\kappa/L}&0\end{bmatrix}\,.$ (S23)

For the filled lower band, we have

$\ket{u_{k}}=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\\
e^{i\phi_{k}}\end{bmatrix},\quad
e^{2i\phi_{k}}=\frac{\lambda+e^{i2\pi\kappa/L}}{\lambda+e^{-i2\pi\kappa/L}},$
(S24)

and

$\ket{\partial_{\lambda}u_{k}}=\frac{\partial_{\lambda}\phi_{k}}{\sqrt{2}}\begin{bmatrix}0\\\
ie^{i\phi_{k}}\end{bmatrix}\,.$ (S25)

We obtain

$F_{\ket{u_{k}}}=2(\partial_{\lambda}\phi_{k})^{2}+4(i\partial_{\lambda}\phi_{k}/2)^{2}=(\partial_{\lambda}\phi_{k})^{2}\,.$
(S26)

At the Dirac point, which is $\kappa=L/2$, we obtain $\phi_{k}=0$ independent
of $\lambda$, and therefore $F_{\ket{u_{N/2}}}=0$. To proceed further, we can
differentiate the expression for $e^{2i\phi_{k}}$ for $\kappa\neq L/2$, which
yields

$\displaystyle
2ie^{2i\phi_{k}}\partial_{\lambda}\phi_{k}=\frac{1}{\lambda+e^{-i2\pi\kappa/L}}-\frac{\lambda+e^{i2\pi\kappa/L}}{(\lambda+e^{-i2\pi\kappa/L})^{2}}=-\frac{2i\sin(2\pi\kappa/L)}{(1+\lambda
e^{-i2\pi\kappa/L})^{2}}\,.$ (S27)

Therefore,

$\partial_{\lambda}\phi_{k}=-\frac{\sin(2\pi\kappa/L)}{1+\lambda^{2}+2\lambda\cos(2\pi\kappa/L)}\,.$
(S28)

At $\lambda=1$, we can further simplify this expression to

$\displaystyle\partial_{\lambda}\phi_{k}$
$\displaystyle=-\frac{\sin(2\pi\kappa/L)}{2\left(1+\cos(2\pi\kappa/L)\right)}=-\frac{\tan(\pi\kappa/L)}{2}\,,$
(S29)

and therefore $F_{\ket{u_{k}}}=\tan^{2}(\pi\kappa/L)/4$. Finally, the QFI of
the many-body ground state is obtained by summing over QFI of all
$\ket{u_{k}}$ except at the Dirac point, which is $\kappa=L/2$. We therefore
get

$\displaystyle F_{\rm PBC}^{\rm SSH}(\lambda_{\rm
c})=\sum_{\kappa=1}^{L-1}\frac{\cot^{2}(\pi\kappa/L)}{4}=\frac{L^{2}-3L+2}{12}\,,$
(S30)

where the last equation is obtained using Mathematica. We therefore have
proven $\Theta(N^{2})$ dependence for the QFI of $\ket{\Psi^{\rm PBC}_{\rm
GS}}$ for the SSH Hamiltonian.

###  I.7 Many-body QFI scaling away from TPT for SSH (PBC)

We have already shown that QFI scales at most linearly away from TPT for the
SSH ground state. In the _continuum_ limit, i.e., $L\rightarrow\infty$, we may
prove this rigorously in the following way. QFI $F_{\rm
PBC}^{\text{SSH}}(\lambda)$ is given by $(\partial_{\lambda}\phi_{k})^{2}$
summed over all modes $\kappa\in[1,L]$, i.e.,

$F_{\rm
PBC}^{\text{SSH}}(\lambda)=\sum_{\kappa=1}^{L}\left(\frac{\sin(2\pi\kappa/L)}{1+\lambda^{2}+2\lambda\cos(2\pi\kappa/L)}\right)^{2}\,.$
(S31)

In the continuum limit, this sum may be approximated by an integral if the
function is Riemann-Integrable. That is,

$F_{\rm
PBC}^{\text{SSH}}(\lambda)=\frac{L}{\pi}\int_{0}^{\frac{\pi}{2}-\epsilon}\left(\frac{\sin
x}{1+\lambda^{2}+2\lambda\cos
x}\right)^{2}dx+\frac{L}{2\pi}\int_{\frac{\pi}{2}+\epsilon}^{2\pi}\left(\frac{2\sin
x}{1+\lambda^{2}+2\lambda\cos x}\right)^{2}dx\,.$ (S32)

Note that we have divided the domain of integration into two discontinuous
domains to reflect the physics at the Dirac Point. It is easy to see that if
this integral exists, then QFI scales linearly with $L$. This is the case away
from TPT, where QFI takes the following form

$\lim_{L\to\infty}\frac{F_{\rm
PBC}^{\text{SSH}}(\lambda)}{L}=\begin{cases}\frac{1}{2(1-\lambda^{2})}&\quad\text{if
}\lambda<1\,,\\\ \frac{1}{2(\lambda^{4}-\lambda^{2})}&\quad\text{if
}\lambda>1\,,\\\ \end{cases}$ (S33)

At TPT, the functions are no longer piecewise Riemann Integrable and linear
scaling no longer applies. This is consistent with the quadratic scaling
derived above.

###  I.8 Many-body QFI scaling at TPT for Chern insulator (PBC)

For the Chern insulator Hamiltonian in Eq. (7) in the main text, the lower-
band eigenvectors are given by,

$\displaystyle\ket{\psi_{\bm{k}}}=\ket{\bm{k}}\ket{u_{\bm{k}}}\,,$ (S34)

with

$\displaystyle\ket{u_{\bm{k}}}=\sqrt{\frac{B_{z}+E_{\bm{k}}}{2E_{\bm{k}}}}\begin{bmatrix}\frac{B_{z}-E_{\bm{k}}}{B_{x}+iB_{y}}\\\
1\end{bmatrix}\,,$ (S35)

where $E_{\bm{k}}=\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}$ is the magnitude of
eigenenergy. After a bit of algebra one finds

$\displaystyle\ket{\partial_{\lambda}u_{\bm{k}}}=\sqrt{\frac{B_{z}+E_{\bm{k}}}{2E_{\bm{k}}}}\frac{E_{\bm{k}}-B_{z}}{2E^{2}_{\bm{k}}}\begin{bmatrix}\frac{B_{z}-E_{\bm{k}}}{B_{x}+iB_{y}}\\\
1\end{bmatrix}\,,$ (S36)

which is orthogonal to $\ket{u_{\bm{k}}}$. Therefore we get

$\displaystyle F_{\rm PBC}^{\rm
Ch}(\lambda)=\displaystyle\sum_{\bm{k}}\braket{\partial_{\lambda}u_{\bm{k}}}{\partial_{\lambda}u_{\bm{k}}}=\displaystyle\sum_{\bm{k}}\frac{B_{x}^{2}+B_{y}^{2}}{4E^{4}_{\bm{k}}}\,.$
(S37)

We use this expression in the main text to extract the scaling of QFI at TPT
and away from it.


# Mechanical classification of voice quality

Akito Yoshida<EMAIL_ADDRESS>Department of Mechano-
Informatics, University of Tokyo, Tokyo 113-8656, Japan  Shigeru Shinomoto
<EMAIL_ADDRESS>Department of Physics, Kyoto University,
Kyoto 606-8502, Japan  Brain Information Communication Research Laboratory
Group, ATR Institute International, Kyoto 619-0288, Japan

###### Abstract

While there is no a priori definition of good singing voices, we tend to make
consistent evaluations of the quality of singing almost instantaneously. Such
an instantaneous evaluation might be based on the sound spectrum that can be
perceived in a short time. Here we devise a Bayesian algorithm that learns to
evaluate the choral proficiency, musical scale, and gender of individual
singers using the sound spectra of singing voices. In particular, the
classification is performed on a set of sound spectral intensities, whose
frequencies are selected by minimizing the Bayes risk. This optimization
allows the algorithm to capture sound frequencies that are essential for each
discrimination task, resulting in a good assessment performance. Experimental
results revealed that a sound duration of about 0.1 sec is sufficient for
determining the choral proficiency and gender of a singer. With a program
constructed on this algorithm, everyone can evaluate choral voices of others
and perform private vocal exercises.

##  I Introduction

A professional assessment of singing performances is based on many criteria
such as musical pitch, rhythm, breathing, and vibrato. However, a lay audience
often makes instantaneous and consistent judgment of the overall quality of
singing without evaluating these criteria. For instance, an audience of a
musical contest can react to every new singing in a few seconds. Such an
instantaneous evaluation might be based on the sound spectrum that can be
perceived in a short time Bartholomew (1934). In particular, it has been
reported that the sound spectra of well-trained male opera singers exhibit a
significant peak at about 3 kHz, which is absent in untrained singers Sundberg
(1990). This spectral peak is called the singer’s formant; it has been
analyzed with spectral modeling synthesis and has been issued as the singing
power ratio Omori  _et al._ (1996); Watts  _et al._ (2006); Aithal  _et al._
(2011); Usha  _et al._ (2017) or quality ratio Manfredi  _et al._ (2015).

Here we considered the sound spectral information and developed a Bayesian
algorithm that learns choral proficiency from sample singing voices. We also
let the algorithm identify the musical scale and whether the owner of the
voice is male or female. The factors useful for each categorization were found
by optimizing the model performance. We also estimated the minimum duration of
a sound required for determining the considered characteristics of individual
singers. By embedding the algorithm into an application program, everyone may
evaluate the choral voice and perform private vocal exercises by oneself,
similar to other applications that evaluate karaoke singing by taking into
account the vocal pitch accuracy Welch  _et al._ (1989); Hoppe  _et al._
(2006); Tsai and Lee (2012); Goto (2014) and vibrato Nakano  _et al._ (2006).

For this purpose, we collected singing voices of subjects as training examples
and categorized them according to whether the subjects had participated in
choral groups (referred to as singers) or not (referred to as non-singers),
and let the algorithm learn this category. Since the participation in choral
groups does not necessarily guarantee good voices, we asked a professional
voice trainer to evaluate the recorded voices and confirmed that our
classification was strongly correlated with the choral voice quality. The
proposed algorithm was trained and evaluated using cross-validation.

##  II Methods

###  II.1 Subjects

We recruited 50 subjects from undergraduate and graduate courses at Kyoto
University (ages ranged from 19 to 27). We classified them into singers and
non-singers according to whether or not they had participated in a mixed
chorus club more than one year. As a result, we acquired singing voices from
23 singers (11 males and 12 females) and 27 non-singers (14 males and 13
females). We asked a professional voice trainer to evaluate their recorded
voices.

###  II.2 Recording

Figure 1: Average power spectra of singers’ voices at every musical scale of
“do re mi fa so la ti do” in C major. The vertical axis represents the
normalized power spectra $I_{\omega}$, while the horizontal axis represents
the frequency $\omega$. The vertical dashed lines represent the basic
frequencies of the eight musical scales. Figure 2: Average power spectra of
the four categories defined as combinations of singer or non-singer and male
or female.

Each subject was asked to sing “a” sound for 1.5 sec by following a piano at
every musical scale of “do re mi fa so la ti do” in C major. The voices were
recorded using a MacBook Air at a sampling rate of 48,000 Hz. We divided the
sound data into intervals of $\Delta=0.1$ sec each and selected 10 pieces with
high intensity. For each piece of sound data, the power spectrum or the
amplitude spectrum is obtained with a fast Fourier transform (FFT). To exclude
the effect of the difference in the sound intensity, we normalized the
original power spectrum $I^{0}_{\omega}$ as

$I_{\omega}=\frac{I^{0}_{\omega}}{\int d\omega I^{0}_{\omega}}.$ (1)

We use the normalized power spectrum $I_{\omega}$ throughout the analysis.

Figure 1 depicts the normalized power spectra of eight musical scales, each
averaged over the singers’ voices. Figure 2 shows four power spectra
classified according to the combinations of singer or non-singer and male or
female.

###  II.3 Inferring voice categories from sound spectrum

We developed a Bayesian method for assessing voice categories (singer or non-
singer and male or female) from a sound spectrum. This can be achieved by
obtaining the conditional distribution $P(c|\\{I_{\omega}\\})$ of a category
or class $c$ given a sound spectrum $\\{I_{\omega}\\}$, which is a function of
the frequency $\\{\omega\\}$. In a practical setting, we divide the frequency
range from 0 to 20 kHz into intervals of 10 Hz. The sound spectrum comprised
2,000 elements. To make the problem tractable, we selected a small number of
power spectral values $\mathbf{x}\equiv$ $\\{I_{\omega_{1}}$,
$I_{\omega_{2}}$, $\cdots$, $I_{\omega_{D}}\\}$ taken at frequencies of
$\\{\omega_{1}$, $\omega_{2}$, $\cdots$, $\omega_{D}\\}$ and represented the
conditional distribution $P(c|\\{I_{\omega}\\})$ as $P(c|\mathbf{x})$. The
frequencies were selected using a modified version of the method for detecting
coins from the colliding sound introduced by one of the authors Furukawa and
Shinomoto (2018). The difference from the original method are summarized
below.

The conditional distribution $P(c|\mathbf{x})$ was obtained with Bayes’
theorem:

$\displaystyle
P(c|\mathbf{x})=\frac{P(\mathbf{x}|c)P(c)}{\sum_{c^{\prime}}P(\mathbf{x}|c^{\prime})P(c^{\prime})},$
(2)

where $P(c)$ denotes the prior distribution of the class $c$. Here we took the
equal prior, $P(c)=1/C$, where $C$ denotes the number of categories (for
instance, $C=2$ for singer vs non-singer). We assumed the class-conditional
density $P(\mathbf{x}|c)$ used in the framework of Gaussian discriminant
analysis Bishop (2006); Murphy (2012):

$\displaystyle
P(\mathbf{x}|c)=\frac{\exp\left\\{-\frac{1}{2}(\mathbf{x}-\bm{\mu}_{c})^{\mathrm{T}}\bm{\Sigma}_{c}^{-1}(\mathbf{x}-\bm{\mu}_{c})\right\\}}{(2\pi)^{D/2}|\bm{\Sigma}_{c}|^{1/2}},$
(3)

where $\bm{\mu_{c}}$ and $\bm{\Sigma_{c}}$ denote the mean and covariance
estimated from the training data, respectively.

For each sound spectrum obtained from the sound data in an interval
$\Delta=0.1$ sec, we estimated the conditional distribution $P(c|\mathbf{x})$
using Eqs. (2) and (3). While each time bin of $\Delta=0.1$ sec gives a fair
inference, integrating the knowledge over 10 intervals may strengthen the
inference. Hence, we used the arithmetic mean of the posterior distributions
$P(c|\mathbf{x})$ computed for 10 intervals. Using the average conditional
distribution, we obtained the maximum a posteriori (MAP) inference as

$\displaystyle c_{\mathbf{x}}=\arg\max_{c}P(c|\mathbf{x}).$ (4)

Given the number of power spectra or dimension $D$, we selected frequencies
$\\{\omega_{1}$, $\omega_{2}$, $\cdots$, $\omega_{D}\\}$ to make the inference
as efficient as possible. This can be achieved by choosing the frequencies
that let the class-conditional densities $\\{P(\mathbf{x}|c)\\}_{c}$ separate
maximally, so that the membership of each sound is optimally distinguished. As
a guiding principle, we suggest minimizing the Bayes risk $R$ or maximizing a
theoretical performance $1-R$ as follows.

$\displaystyle 1-R=\int
d\mathbf{x}\,P(c_{\mathbf{x}}|\mathbf{x})P(\mathbf{x})=\int
d\mathbf{x}\,P(\mathbf{x}|c_{\mathbf{x}})P(c_{\mathbf{x}}).$ (5)

We explored a set of frequencies $\\{\omega_{1}$, $\omega_{2}$, $\cdots$,
$\omega_{D}\\}$ by readjusting every single frequency $\omega_{i}$ to minimize
the Bayes risk $R$ sequentially from $i=1$ to $D$, similar to Algorithm 1 in
Furukawa and Shinomoto (2018). In the present study, we searched the space of
$\log\omega$ instead of real $\omega$ because the important frequencies are
distributed in the range lower than that of inferring coins. To perform the
search in the logarithmic scale, we re-sampled the power spectrum in the
$\log\omega$ axis by interpolating $\log I_{\omega}$.

##  III Results

###  III.1 Inferring musical scales

Figure 3: Cross-validated performance of the proposed algorithm in identifying
musical scales. The performance achieved by selecting frequencies using Bayes
risk minimization is plotted against the performance obtained using randomly
chosen frequencies. Figure 4: Selected frequencies and power spectra. (a)
Frequencies selected by minimizing the Bayes risk for each dimensionality $D$.
(b) Power spectra of eight musical scale voices averaged over all singers.
Dashed lines indicate the selected frequencies for $D=8$.

First, we used the proposed algorithm to infer the musical scale of a voice,
assigning it to match one of the “do re mi fa so la ti do” in C major. In this
experiment, we used voices of 23 singers, who are expected to sing on the key.
The predictive performance was estimated using cross-validation; we repeat the
testing over the voices of randomly chosen 5 subjects using the algorithm
trained on the remaining 18 subjects.

We found that the performance of identifying musical scales increases on
average with the number of parameters or dimension $D$. It can be noticed from
Fig. 3 that the performance is much higher for the case of selecting
frequencies $\\{\omega_{1}$, $\omega_{2}$, $\cdots$, $\omega_{D}\\}$ by
minimizing the Bayes risk compared to the case of randomly selected
frequencies from the uniform distribution in $\log\omega$.

Figure 4(a) shows the frequencies $\\{\hat{\omega}_{1}$, $\hat{\omega}_{2}$,
$\cdots$, $\hat{\omega}_{D}\\}$ selected for the entire data set by minimizing
the Bayes risk $R$ for various dimensions $D$. The frequencies selected for
different dimensions $D$ tend to be similar to each other. By plotting the
power spectra of the eight musical scales, we can see that the selected
frequencies mostly coincide with the basic frequencies of the musical scales
(Fig. 4(b)). This indicates that frequencies can be selected reasonably by
simply following the principle of the Bayes risk minimization.

###  III.2 Inferring male or female

Figure 5:  (a) Performances of the algorithm in distinguishing between males
and females based on their voices. (b) Selected frequencies and power spectra.
Frequencies selected by minimizing the Bayes risk, and the averaged power
spectra for males and females.

Next, we used the algorithm to identify the gender of singers from their
voices. The predictive performance was estimated using cross-validation; we
repeat the testing of 5 male and 5 female voices randomly chosen from 25 males
and 25 females, using the algorithm trained with the remaining 40 subjects.

The performance of identifying the gender of singers was almost perfect for
$D\geq 2$, when using the optimized frequencies (Fig. 5(a)). The frequencies
selected by minimizing the Bayes risk mostly ranged from 100 to 250 Hz (Fig.
5(b)), implying that the algorithm made use of the differences in the power in
the lower tones; males were producing lower harmonics for each given key
compared to females.

###  III.3 Inferring singer or non-singer

Figure 6:  (a) Performances of the algorithm in discriminating singers and
non-singers. (b) Selected frequencies and power spectra. Frequencies selected
by minimizing the Bayes risk and the averaged power spectra for males and
females.  Figure 7:  (a) Power spectra of singers and non-singers of males and
selected frequencies. (b) Power spectra of singers and non-singers of females
and selected frequencies.

The predictive performance of the algorithm for distinguishing between singers
and non-singers was estimated using cross-validation. In particular, we
repeated the testing of 5 singer and 5 non-singer voices randomly chosen from
23 singers and 27 non-singers, respectively, using the algorithm trained with
the remaining 40 subjects.

The predictive performance was improved by selecting frequencies and became
more than 80% correct (Fig. 6(a)). Several specific frequencies were selected
by minimizing the Bayes risk (Fig. 6(b)), including the singer’s formant at
about 3 kHz.

The predictive performance of distinguishing between singers and non-singers
became higher (about 90 %) when we limited the population to only males.
Namely, we estimated the model performance by testing voices of 3 singers and
3 non-singers randomly chosen from 11 male singers and 14 male non-singers,
using the algorithm trained with the remaining 19 subjects. In this case, the
singer’s formant at about 3 kHz was more prominent (Fig. 7(a)). In contrast,
the predictive performance of distinguishing between singers and non-singers
was lower (about 70 %) when we limited the population to only females. In this
case, there was no prominent singer’s formant at 3kHz; interestingly however,
there was a small peak at about 10 kHz (Fig. 7(b)).

###  III.4 Inferring male or female and singer or non-singer

Figure 8:  Inferring singer or non-singer (S/N) and male or female (M/F) in
different order. (a) Infer S/N and then infer M/F,
$P(\textrm{M/F}|\textrm{S/N},\mathbf{x})P(\textrm{S/N}|\mathbf{x})$. (b) Infer
M/F and then infer S/N,
$P(\textrm{S/N}|\textrm{M/F},\mathbf{x})P(\textrm{M/F}|\mathbf{x})$.  Figure
9: Performance of the four methods for capturing M/F and S/N categories.

There are four ways of capturing two types of information (male or female
(M/F) and singer or non-singer (S/N)) from the sound spectrum or
$\mathbf{x}\equiv$ $\\{I_{\omega_{1}}$, $I_{\omega_{2}}$, $\cdots$,
$I_{\omega_{D}}\\}$ as described below.

  * •

Infer M/F and S/N independently:

$P(\textrm{M/F}|\mathbf{x})P(\textrm{S/N}|\mathbf{x})$

  * •

Infer S/N and then infer M/F (Fig. 8(a)):

$P(\textrm{M/F}|\textrm{S/N},\mathbf{x})P(\textrm{S/N}|\mathbf{x})$

  * •

Infer M/F and then infer S/N (Fig. 8(b)):

$P(\textrm{S/N}|\textrm{M/F},\mathbf{x})P(\textrm{M/F}|\mathbf{x})$

  * •

Infer M/F and S/N simultaneously:

$P(\textrm{M/F},\textrm{S/N}|\mathbf{x})$

We compared the performance of these four types of inference in Fig. 9. Here,
a set of frequencies is optimized in each dimensionality $D$. The performance
for the case of inferring M/F and S/N simultaneously is slightly lower
compared to that of the other three methods.

###  III.5 Comparison between the evaluations made by the algorithm and a
professional voice trainer

Figure 10:  Comparison between the evaluations made by the trained algorithm
and a professional voice trainer. The vertical axis represents the posterior
distribution of the Bayesian estimator for the singer or non-singer categories
$P(\textrm{S/N}|\mathbf{x})$.

Since the participation in a choral group does not necessarily guarantee good
voices, we asked a professional voice trainer to evaluate the recorded voices.
By comparing the posterior distribution of the Bayesian estimator
$P(\textrm{S/N}|\mathbf{x})$ with the evaluation score provided by the
professional voice trainer, we confirmed that they are strongly correlated
(Fig. 10).

###  III.6 Duration of a sound required for the estimation

Figure 11:  Performance of inference vs sound length. (a) Inference of the
male and female (M/F) categories. (b) Inference of the singer and non-singer
(S/N) categories.

To estimate the minimum duration of a sound required for determining
characteristics such as male or female and singer or non-singer, we estimated
the predictive performance of the algorithm by shortening the duration of
analysis $10\times\Delta$ from 1 sec. It can be observed from Fig. 11 that the
performance is mostly stable at $10\times\Delta=0.1$ sec, implying that the
inference of male or female and evaluation of singing voices can be performed
almost instantaneously.

##  IV Discussion

The professional assessment of singing is a complicated task accounting for
various factors such as musical pitch, rhythm, breathing, and vibrato. There
have been attempts to achieve the automatic evaluation of singing by
considering these factors. For instance, Nakano et al. presented an automated
algorithm for evaluating the singing skill by detecting the pitch interval
accuracy and vibrato Nakano  _et al._ (2006). There are studies utilizing
neural network learning that aim to achieve high performance in the assessment
Böhm  _et al._ (2017); Vidwans  _et al._ (2017); Pati  _et al._ (2018); Wu and
Lerch (2018); Zhang  _et al._ (2019).

In the present study, we have developed an algorithm for assessing music
performances based solely on the sound spectrum of singing voices. The
algorithm can evaluate the choral proficiency of a singer fairly well without
accounting for the time variability of songs. In particular, our algorithm was
able to make a reasonable assessment from a short passage of a singing voice
of about 0.1 sec, with the information of rhythm, breathing, and vibrato being
unavailable. This is similar to audiences making instantaneous voice
evaluations when hearing new singing performances in musical contests.

For each task of the categorization regarding the choral proficiency, musical
scale, and gender of a singer, our learning algorithm selects frequencies of
the sound spectrum by minimizing the Bayes risk. This optimization allows the
algorithm to capture sound frequencies that are essential from each
discrimination task, resulting in a good assessment performance. In
particular, frequencies close to the basic frequencies or their harmonics were
selected to separate musical scales; low frequencies of 100-300 Hz, at which
the male voices have stronger power, were selected for separating males and
females; and frequencies of about 3 kHz and 10 kHz were selected for males and
females, respectively, when separating singers and non-singers. The good
performance achieved in distinguishing between male singers and non-singers
may be largely due to the detection of singer’s formant at 3 kHz. The good
performance in categorizing female singers and non-singers may be largely due
to the peak at 10 kHz. It should be noted that the categorization of the
female singing proficiency was not as good as that achieved for males.
Regarding the duration of hearing a sound, it was established that 0.1 sec is
enough to distinguish between males and females or singers and non-singers.

When analyzing the singing proficiency, we found the singer’s formant or
prominent peak of the power spectrum to be at 3 kHz for male singers and 10
kHz for female singers. We were interested to find out whether these peaks are
the cause of good singing voices or epiphenomena of good voices. To
investigate this, we generated colored noises with frequencies distributed at
about 3kHz or 10 kHz, added them into the singing voices, and asked the
professional voice trainer to evaluate the sounds. As a result, we did not
observe significant changes in the evaluation of good voices. Nevertheless,
the current experimental setup was not perfect for answering this question and
we plan to investigate it in more detail in the future.

In the present study, we focused on proficiency in classical choral singing.
Since the method of Bayesian learning presented in this study is a general
framework applicable to any sounds, it would be interesting to apply the
method to other styles of singing, such as pop, soul, country, folk, and
metal, and see the difference in the important frequencies for evaluating the
proficiency of singing in these styles.

## ACKNOWLEDGMENTS

We thank Kazuki Nakamura and Sou Shinomoto for collecting voices, and Kinya
Nakanishi for evaluating singing voices. S.S. is supported by JST CREST Grant
Number JPMJCR1304, and the New Energy and Industrial Technology Development
Organization (NEDO).

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