# Error Mitigation Thresholds in Noisy Quantum Circuits

Pradeep Niroula  Joint Center for Quantum Information and Computer Science,
NIST/University of Maryland, College Park, Maryland 20742, USA  Sarang
Gopalakrishnan  Department of Electrical Engineering, Princeton University,
Princeton, NJ 08544, USA.  Michael J. Gullans  Joint Center for Quantum
Information and Computer Science, NIST/University of Maryland, College Park,
Maryland 20742, USA

###### Abstract

Extracting useful information from noisy near-term quantum simulations
requires error mitigation strategies. A broad class of these strategies rely
on precise characterization of the noise source. We study the performance of
such strategies when the noise is imperfectly characterized. We adapt an Imry-
Ma argument to predict the existence of an _error mitigation threshold_ for
random spatially local circuits in spatial dimensions $D\geq 2$:
characterization disorder below the threshold rate allows for error mitigation
up to times that scale with the number of qubits. For one-dimensional
circuits, by contrast, mitigation fails at an $\mathcal{O}(1)$ time for any
imperfection in the characterization of disorder. We discuss implications for
tests of quantum computational advantage, fault-tolerant probes of
measurement-induced phase transitions, and quantum algorithms in near-term
devices.

Noise presents a fundamental barrier to realizing scalable quantum information
processing Nielsen and Chuang (2011). The theory of fault-tolerant quantum
error correction shows how this barrier can be overcome in principle Aharonov
and Ben-Or (1997); Dennis  _et al._ (2002); Gottesman (2009). However, despite
remarkable experimental progress establishing the basic validity of the theory
of fault-tolerance Egan  _et al._ (2021); Krinner  _et al._ (2022); Acharya
_et al._ (2022); Sivak  _et al._ (2022), achieving large-scale quantum
computing with error corrected qubits remains a formidable challenge. In
recent years, quantum error mitigation has emerged as a complementary paradigm
for addressing the effects of noise in large-scale quantum devices McArdle
_et al._ (2020); Cai  _et al._ (2022); Temme  _et al._ (2017); Li and Benjamin
(2017); Kim  _et al._ (2023); McClean  _et al._ (2017); Bonet-Monroig  _et
al._ (2018); McArdle  _et al._ (2019); Cotler  _et al._ (2019); Huggins  _et
al._ (2021); Koczor (2021); Czarnik  _et al._ (2021); Strikis  _et al._
(2021). At its core, error mitigation relies on the ability to accurately
characterize the interaction of the system with its environment. Armed with
this knowledge, one can design classical post-processing techniques to
construct more accurate estimators of the noiseless signal using data obtained
from a noisy device. Similar methods have been ubiquitously employed in
quantum process tomography to reliably extract error models in the presence of
noisy operations Emerson  _et al._ (2005); Knill  _et al._ (2008); Nielsen
_et al._ (2021); using these methods to improve the accuracy of noisy quantum
circuits is a more recent development.

A key aspect of the threshold theorem for fault-tolerant quantum computing is
that it applies even in the case where the individual components used to
implement the error correction are noisy. An analogous question for error
mitigation strategies is whether they work when the noise is imperfectly
characterized. As noted above, related questions have been addressed for
quantum process tomography in the presence of faulty operations Emerson  _et
al._ (2005); Knill  _et al._ (2008); Nielsen  _et al._ (2021); however, these
models are often considered only at the level of a few qubits, generally
precluding the existence of a sharply defined phase transition. In the case of
many-body tomographic problems (e.g., Pauli noise estimation Fla ; Harper  _et
al._ (2020) or Hamiltonian learning Bairey  _et al._ (2019); Anshu  _et al._
(2021)), thresholds in learnability are possible and likely arise under some
circumstances. However, to our knowledge, threshold results have not been
considered for quantum error mitigation.

In this paper, we demonstrate the possibility of thresholds for quantum error
mitigation. We focus on one of the conceptually simplest quantum error
mitigation techniques, which goes under the moniker “probabilistic error
cancellation” (PEC) Temme  _et al._ (2017); Berg  _et al._ (2022). PEC relies
on the fact that many of the most common noise channels used to model quantum
devices are actually trivially invertible at the mathematical level.
Unfortunately, the inverse operation is not generally a physical evolution.
For example, the inverse of the depolarizing channel violates the complete-
positivity condition for quantum channels. As a result, the only known way to
implement the inverse is through classical post-processing. In the case of
PEC, this classical post-processing step is implemented via Monte Carlo
sampling methods but, for constant noise rates, incurs exponential sampling
overheads in the worst case Temme  _et al._ (2017); Berg  _et al._ (2022).
Nevertheless, PEC can be used to significantly extend the accessible circuit
depths of a given noisy quantum device, as demonstrated in recent experiments
Berg  _et al._ (2022). Some other examples of quantum error mitigation include
zero-noise extrapolation Temme  _et al._ (2017); Li and Benjamin (2017); Kim
_et al._ (2023), symmetry-based error detection Bonet-Monroig  _et al._
(2018); McArdle  _et al._ (2019), cooling/purification Cotler  _et al._
(2019); Huggins  _et al._ (2021); Koczor (2021), and learning based methods
Czarnik  _et al._ (2021); Strikis  _et al._ (2021). In the remainder of the
paper we will focus on the PEC technique, primarily because of its conceptual
simplicity.

Our central prediction is that PEC has an error mitigation threshold for
random, spatially local circuits in spatial dimensions $D\geq 2$ when the
noise is imperfectly characterized. Building on the extensive work
characterizing quantum phases and phase transitions in hybrid random circuits
Fisher  _et al._ (2022); Potter and Vasseur (2022), we consider a model of
random unitary circuits subject to depolarizing noise with binary disorder in
the depolarization rate [see Fig. 1(a)]. To mitigate the effects of the noise
we apply a uniform “antinoise” channel that inverts the depolarizing noise on
average. We show that a replica statistical mechanics description of the
problem has close parallels to the classical random field Ising model (RFIM)
in $D+1$ dimensions. At the zero-mean field condition in the RFIM, Imry-Ma
arguments Imry and Ma (1975) show that the ordered phase survives random
symmetry-breaking terms in sufficiently high dimensions. In the random circuit
problem, this analysis indicates the possibility for error mitigation
thresholds for $D\geq 2$, while $D=1$ is the marginal dimension. We also
analyze the performance of error mitigation as a function of circuit depth.
For depths larger than the linear size of the system, the mitigation fails,
resulting in a signal worse than that obtained by sampling from the uniform
distribution. The corresponding phase diagram of the system is shown in Fig.
1(b) for $D\geq 2$. Analytic and numerical studies of the two-copy replica
theory and exact simulations of the mitigated circuit dynamics confirm our
predictions.

Figure 1: Model and phase diagram: (a) Noisy random circuit with spacetime
disorder in the noise rate and error mitigation. (b) Phase diagram for the
error mitigation threshold in this model for $D\geq 2$ spatial dimensions. At
high disorder rates and linear depths, the system transitions from a “below-
threshold” phase where mitigation succeeds on average to an “above-threshold”
phase where mitigation fails.

_Model and Simplified Limits.—_ The basic model we consider in this work is
illustrated in Fig. 1(a). Each two-site gate is chosen to be Haar random Nahum
_et al._ (2017, 2018); von Keyserlingk  _et al._ (2018) and are interspersed
with noise and antinoise channels. We focus on the case of depolarizing noise
and its inverse channel as the antinoise

$\displaystyle\mathcal{E}_{\bm{x},t}(\rho)$
$\displaystyle=(1-q_{\bm{x},t})\rho+q_{\bm{x},t}\operatorname{tr}_{\bm{x}}[\rho]\otimes
I/2,$ (1) $\displaystyle\mathcal{A}_{\bm{x}}(\rho)$ $\displaystyle=\frac{\rho-
q_{a}\operatorname{tr}_{\bm{x}}[\rho]\otimes I/2}{1-q_{a}},$ (2)

where $\operatorname{tr}_{\bm{x}}[\rho]$ is a partial trace operation at site
$\bm{x}$. Tuning $q_{a}=q_{\bm{x},t}$ inverts the noise at space-time site
$(\bm{x},t)$ such that
$\mathcal{A}_{\bm{x}}\circ\mathcal{E}_{\bm{x},t}(\rho)=\rho$. To include the
disorder, the noise rates are drawn randomly at each space-time location of
the circuit from one of two values $q_{\bm{x},t}=q_{1/2}$ with probability
$p/1-p$ and $q_{1}<q_{2}$. The antinoise can be tuned to a zero mean-field
condition by taking $(1-q_{a})=(1-q_{1})^{p}(1-q_{2})^{1-p}$ equal to a
geometric mean. This condition ensures that a Pauli operator
$\sigma_{\bm{x}}^{\mu}$ ($\mu\in\\{X,Y,Z\\}$) evolved under multiple rounds of
noise and antinoise up to depth $t=d$ will have expectation value

$\operatorname{tr}\bigg{[}\sigma_{\bm{x}}^{\mu}\prod_{t=1}^{d}\mathcal{A}_{\bm{x}}\circ\mathcal{E}_{\bm{x},t}(\rho)\bigg{]}=\prod_{t}\frac{1-q_{\bm{x},t}}{1-q_{a}}\operatorname{tr}[\sigma_{\bm{x}}^{\mu}\rho],$
(3)

where the prefactor tends to a log-normal distribution that concentrates
around one. Any deviations of $q_{a}$ from this zero mean-field condition will
result in an exponential growth rate that scales linearly in the depth $d$. At
the zero-mean field condition, however, the noise is canceled on average and
the prefactor deviates from one only due to random fluctuations as
$e^{\pm\mathcal{O}(\sqrt{d})}$ (i.e., the exponential growth rate is
suppressed by a factor $\sqrt{d}$).

To develop some basic intuition for how unitary dynamics can improve the
robustness of error mitigation, it is helpful to consider a toy model
consisting of two sites $L/R$ with $q_{L}<q_{R}$ fixed in time. Now, take
$1-q_{a}=\sqrt{(1-q_{R})(1-q_{L})}$ to satisfy the zero mean-field condition.
The $R$ site will be subject to excess noise and the $L$ site will experience
excess antinoise. The case of weak unitary dynamics is analogous to the above
threshold phase. In the limit where the unitary dynamics tends to trivial
evolution, any state initialized with non-zero Pauli expectation value on site
$L$ will experience exponential growth of the expectation values leading to an
unphysical density matrix after an $\mathcal{O}(1)$ time. We discuss below how
this type of rapid instability to unphysical states is characteristic of the
above threshold phase.

To build in the effects of unitary dynamics, consider the base circuit to
consist of repeated applications of SWAP gates. In this case, it is readily
verified that the effects of the noise on Pauli expectation values will be
completely canceled for arbitrarily deep circuits except for the most recent
layer of gates. In this two-qubit example, Haar random gates behaves similarly
to SWAP gates in that the noise is cancelled on average. However, the temporal
randomness will lead to exponential fluctuations of Pauli expectation values
of size $e^{\mathcal{O}(\sqrt{d})}$, where the term in the exponent will
follow the statistics of a random walk. This tendency of the random unitary
dynamics to suppress effects of the noise/antinoise from linear in $d$ to
$\sqrt{d}$ scaling for certain initial conditions is characteristic of the
behavior we find for the below-threshold phase. To more quantitatively capture
the trends found in this simple two-site model, we now develop a mean-field
theory for the threshold.

_Mean-field theory.—_ In deriving a mean-field theory, it is convenient to
turn to a more analytically tractable model formulated in continuous time
rather than as a discrete circuit. The noise and antinoise are now captured
through a Lindblad master equation Sun  _et al._ (2021)

$\displaystyle\dot{\rho}$
$\displaystyle=-i[H(t),\rho]-\sum_{j}(\gamma_{j}-\gamma_{a})\big{(}\rho-\operatorname{tr}_{j}[\rho]\otimes
I/2\big{)},$ (4) $\displaystyle H$
$\displaystyle(t)=\sum_{i<j,\mu,\nu}J_{ij\mu\nu}(t)\sigma_{i}^{\mu}\sigma_{j}^{\nu},$
(5)

where $H(t)$ is a time-dependent Hamiltonian that sets the unitary evolution,
$\gamma_{i}$ is the random noise rate on site $i$, and $\gamma_{a}$ is the
antinoise rate. The noise rate $\gamma_{i}=\gamma_{1/2}$ is drawn randomly
with probability $p/1-p$. The zero mean-field condition becomes
$\gamma_{a}=\bar{\gamma}\equiv p\,\gamma_{1}+(1-p)\,\gamma_{2}$. We take the
random unitary dynamics to be given by an all-to-all coupled Brownian circuit
model in which the $J_{ij\mu\nu}(t)$ are drawn from a white-noise Gaussian
random process with variance parameters Lashkari  _et al._ (2013); Jian  _et
al._ (2022)

$\langle
J_{ij\mu\nu}(t)J_{k\ell\gamma\delta}(t^{\prime})\rangle=\frac{J}{2N}\delta_{ik}\delta_{j\ell}\delta_{\mu\gamma}\delta_{\nu\delta}\delta(t-t^{\prime}).$
(6)

To treat this model analytically, we average the dynamics over the random
coupling constants for multiple identical copies of the density matrix
$\rho(t)^{\otimes k}$ ($k$ is a replica index). In the equations of motion for
the averaged moments $\rho_{k}(t)\equiv\mathbb{E}_{H}[\rho^{\otimes k}]$, the
unitary drive-term averages to zero, leading to a purely dissipative master
equation for $\rho_{k}(t)$ (see Appendix A). Focusing on second moments
$(k=2)$ gives rise to a particularly simple model where with two global steady
states given by $I^{\otimes N}$ and $S^{\otimes N}$ Jian  _et al._ (2022),
where $S$ is the SWAP operator acting on the two copies. The all SWAP steady-
state captures non-trivial corrections to second moment observables like the
purity of sub-systems. In the mean-field approximation, we enforce the density
matrix to take a product state form $\rho_{2}=\bigotimes_{i=1}^{N}\sigma_{i}$.
Due to the lack of symmetries or conservation laws in the problem,
$\sigma_{i}$ can be parameterized as

$\sigma_{i}=(1/4+\delta_{i})|s\rangle\langle s|+(1/4-\delta_{i}/3)P_{T},$ (7)

where $\delta_{i}$ is the deviation from an infinite temperature state,
$|s\rangle$ is a two-qubit singlet state across the two copies, and $P_{T}$ is
the projector onto the two-qubit triplet subspace of the two copies.

The mean-field equations for $\delta_{i}$ take the form

$\dot{\delta}_{i}=-4\Big{[}\Delta_{i}+\frac{J}{N}\sum_{j\neq
i}(3+4\delta_{j})\Big{]}\delta_{i},$ (8)

where $\Delta_{i}=\gamma_{i}-\gamma_{a}$. This simple equation captures much
of the basic physics we described in the two-site model, while also exhibiting
the phase transition. Notably, we can see that at weak values of the disorder,
the interaction term acts as a restoring force towards the fixed point
$\delta_{i}=0$. The other fixed point $\delta_{i}=-3/4$ corresponds to an
unphysical density matrix, which is unstable at weak disorder. At a critical
disorder strength of $|\Delta_{1}|=3J$ (see App. A), however, the
$\delta_{i}=0$ fixed point becomes unstable. The dynamics flows to a new
stable fixed point which is an unphysical density matrix. Thus, we see that
the above threshold phase in the mean-field theory is characterized by an
instabilty of the density matrix to unphysical states.

To rigorously characterize the threshold behavior, particularly in finite-
dimensional systems, we need a more systematic treatment of the correlations
in the state beyond the mean-field approximation. To develop this approach, it
is convenient to return to the circuit models. In these discrete models, the
connections to the physics of the RFIM become more explicit.

_Statistical Mechanics Mapping.—_ Our rigorous analysis of the problem relies
on well-established mappings from random quantum circuit dynamics in $D$
spatial dimensions to statistical mechanics models in $D+1$ dimensions Nahum
_et al._ (2018); von Keyserlingk  _et al._ (2018); Jian  _et al._ (2020); Bao
_et al._ (2020); Dalzell  _et al._ (2022, 2021); Deshpande  _et al._ (2022);
Gao  _et al._ (2021). We outline the details of the mapping in Appendix B. The
approach we take is to expand $\rho_{k}\equiv\mathbb{U}[\rho^{\otimes k}]$
into a basis of operators based on representations of the permutation group
$S_{k}$ Dalzell  _et al._ (2022). One can then derive update rules for the
averaged state following each layer of gates, noise, and antinoise. In this
paper, we focus on the two-replica case where the two operators are $I$ and
$S$ (the SWAP gate). We expect this model to be sufficient to capture many of
the properties of the transition, as has been argued in the unitary case Nahum
_et al._ (2018). Numerical evidence based on exact simulations is presented
below that support this claim.

Without noise/antinoise, $\rho_{2}$ has two fixed points $I^{\otimes N}$ and
$S^{\otimes N}$ just as in the Brownian circuit model. Configurations with
persistent “domain walls” (i.e., mixtures of $I$ and $S$) are exponentially
damped with depth in the model. As a result, the two fixed points can be
interpreted as ferromagnetic ground states of an Ising-type spin model. In
this interpretation, noise acts as a symmetry breaking field because it favors
the identity configuration and damps out configurations with $S$ operators Bao
_et al._ (2020). antinoise acts as a field in the opposite direction, but it
also adds additional sign-structure to the problem from the second term in Eq.
(2). We account for this sign by labeling spacetime configurations of $I$ and
$S$ operators by the sign of their contribution, then we can write
$\rho_{2}=\rho_{2}^{+}-\rho_{2}^{-}$, where $\rho_{2}^{\pm}$ sum over
spacetime configurations with a positive/negative sign.

Figure 2: Numerical tests of the threshold theory: (a) Numerical simulations
of circuit and noise-disorder averaged the correlation metric $\tilde{I}_{12}$
obtained from purity of subregions for two random sites in the system. (b)
Mutual information $I_{12}$ for exact simulations of the mitigated noisy
random circuit. (c-b) Scaling collapse of data from (a-b) using
$\sigma_{c}/\bar{q}=0.65(5)/0.55(5)$ and critical exponent on $x$-axis
obtained from local probe (see Appendix C). The critical scaling on the
$y$-axis is an ansatz. (e-f) Similar quantities as (a-b) for antipodal sites
of a 1D chain. The strong drift in the peak with increasing system size is
consistent with the lack of a below-threshold phase in this model.

_Imry-Ma arguments.—_ To make precise connections to Imry-Ma arguments Imry
and Ma (1975), we rigorously analyze finite-dimensional versions of the
statistical mechanics models. First, we define a simple initial condition as a
state $\rho_{2}$ that is identity everywhere except in a contiguous region $A$
where it is given by $\operatorname{tr}_{A^{c}}[\rho_{2}]\propto
I^{\otimes|A|}+S^{\otimes|A|}$. Such an initial condition can be prepared
physically by taking a Haar random state (or a state drawn from a two-design
such as the Clifford group DiVincenzo  _et al._ (2 03)) on $A$ and the
infinite temperature state on the complement of $A$. The first model we
consider is a _quenched 1D_ model described by a one-dimensional brickwork
circuit of the type shown in Fig. 1(a) with Haar random two-qubit gates and
spatially random noise rates that are constant in time. We define
$\sigma=\sqrt{p(1-p)(q_{2}-q_{1})}$ as the standard deviation of the noise
rates. We prove that the quenched 1D model always exhibits an instability for
one of the simple initial conditions.

######  Theorem 1 (Instability in quenched 1D).

For the quenched 1D model with any $\sigma>0$, there is a simple initial
condition for subregion $A$ of size $\mathcal{O}(\log N)$ such that as
$N\to\infty$ and $d\to\infty$ with high probability $||\rho_{2}^{\pm}||\geq
e^{\mathcal{O}(d\log N)}/2^{2N-|A|}3^{|A|}$.

###### Proof.

The proof of the theorem essentially follows a standard Imry-Ma argument. For
any finite $\sigma$ as $N\to\infty$ with high probability (whp) there will be
a contiguous region of sites $A$ with noise rate $q_{1}$ of size
$|A|=\mathcal{O}(\log N)$. For sufficiently large $N$ there will be a region
satisfying $(1-q_{1})^{|A|}/(1-q_{a})^{|A|}>(5/2)^{2}$ whp, which implies that
the $\mathcal{O}(1)$ boundary cost of the two domain walls at the edge of $A$
can be overcome by the amplification of the $S$ configurations in the bulk.
Taking a configuration of all $S$ on $A$ with identities elsewhere that moves
by order 1 site at the boundary at each time-step according to the rules for
the replica model provides a contribution to $\rho_{2}^{\pm}$ that diverges as
$e^{\mathcal{O}(d\log N)}/2^{2N-|A|}3^{|A|}$. The denominator is the
normalization constant for the initial state. ∎

As a corollary to this result, an initial pure product state evolved with this
model will also have this instability. The converse of this type of Imry-Ma
argument is that we expect the instability of these initial conditions to be
suppressed for weak enough disorder for high enough dimensions. In this case,
for large regions $A$ the boundary cost of the domain wall scales as
$|A|^{(D-1)/D}$, whereas the amplification from the antinoise scales as
$|A|^{1/2}$. As a result, with quenched disorder in time, we expect the
instability to vanish for $D\geq 3$ at sufficiently small values of $\sigma$,
while $D=2$ is the marginal dimension. For space-time random disorder, the
marginal dimension is $D=1$ and an ordered phase persists for $D\geq 2$. Note,
that at large values of $\sigma$, the domain wall cost can be overcome even by
a single site with strong antinoise, leading to an instability. As a result,
there will be a disorder driven phase transition at finite $\sigma$ in
sufficiently high-dimensions associated with the onset of this instability.

_Numerical Tests.—_ Figure 2 summarizes our numerical results. We run
numerical simulations of random circuits, together with noise and antinoise,
to characterize this phase transition in two settings. In the first case [see
Fig. 2(a-d)], we simulate an infinite-dimensional circuit where entangling
gates can exist between any pair of qubits and noise is random in both space
and time. In the second case [see Fig. 2(e-f)], we simulate $(1+1)D$ circuit
dynamics assuming quenched noise, where the randomness is only spatial. The
first model is a test of the high-dimensional limit of the model, while the
second model is the same one considered in Theorem 1.

In both cases, we initialize a state of a $N$-qubit register in a global Haar-
random state, which corresponds to the string $\propto I^{\otimes
N}+S^{\otimes N}$ in the statistical-mechanical mapping. For the infinite-
dimension model, we apply random Haar gates on a set of $N/2$ disjoint pairs,
randomly drawn from all $N(N-1)/2$ possible pairs of qubits. For the $(1+1)D$
model, each time-step corresponds to a layer in the brickwork circuit in Fig.
1 with periodic boundary conditions. Each layer of unitary gates is followed
by the noise-antinoise channel on all qubits.

After evolving the circuit to depth $d$, we take the partial trace of two
qubits and calculate correlation metrics between the two qubits Skinner  _et
al._ (2019); Li  _et al._ (2019). These low-order correlations are
exponentially suppressed in $N$ for $\sigma=0$, but they develop a peak at the
critical point where long-range correlations develop. For $D=1$, the pair
consists a randomly chosen qubit and its corresponding anti-pode at a distance
of $N/2$; for the all-to-all model, both qubits are randomly chosen.

We study the two systems using the replica statistical-mechanical mapping
introduced above, with background details in B. The mapping lets us probe the
phase transition for system sizes as large as $N=20$. For sizes up to $N=10$,
we complement the numerics on statistical-mechanical mapping with direct
circuit simulations. In the replica stat-mech model we study a measure of
correlations
$\tilde{I}_{ab}=-\log_{2}\mathbb{E}_{U}[\mathrm{Tr}[\rho_{a}^{2}]]-\log_{2}\mathbb{E}_{U}[\mathrm{Tr}[\rho_{b}^{2}]]+\log_{2}\mathbb{E}_{U}[\mathrm{Tr}[\rho_{ab}^{2}]]$
averaged over noise-disorder (denoted by $\langle\cdot\rangle$ brackets),
while in the exact simulations we study the standard mutual information
$I_{ab}=S_{a}+S_{b}-S_{ab}$ for the von Neumann entropy
$S_{a}=-\operatorname{tr}[\rho_{a}\log_{2}\rho_{a}]$ Nielsen and Chuang
(2011).

The numerical results provide strong supporting evidence for the theoretical
scenario outlined above. We see evidence for a stable ordered phase in the
high-dimensional limit of the all-to-all model, whereas the 1D model with
quenched randomness in the noise is consistent with the lack of an ordered
phase. In addition, we estimate critical exponents for the transition in Fig.
2(b-c) that are consistent with simple rational values characteristic of mean-
field behavior.

_Outlook.—_ An important application of our results is to improved
benchmarking and verification of random circuit sampling in noisy devices
Boixo  _et al._ (2018); Arute  _et al._ (2019); Bouland  _et al._ (2019);
Movassagh (2019); Wu  _et al._ (2021); Zhu  _et al._ (2021). In Appendix D, we
demonstrate an exponential improvement in a mitigated version of the so-called
“linear cross-entropy” fidelity below the error mitigation threshold. Notably,
this mitigated benchmark can be computed using existing experimental data. The
below threshold behavior may have important consequences for proofs of quantum
computational advantage in noisy circuits Aaronson and Gunn (2019); Bouland
_et al._ (2021); Deshpande  _et al._ (2022), potentially motivating the
inclusion of error mitigation in complexity theoretic arguments. Moreover, the
exponential improvement in the fidelity score for $D\geq 2$ might indicate a
method to foil “spoofing” algorithms that have been developed for unmitigated
fidelity benchmarks in noisy random circuits Gao  _et al._ (2021).

These results also have implications for measurement-induced phase transitions
in hybrid random circuits Skinner  _et al._ (2019); Li  _et al._ (2019);
Fisher  _et al._ (2022); Potter and Vasseur (2022). Such models are not fault-
tolerant because circuit-level depolarizing noise drives the system to a
short-range correlated mixed state (it acts as a symmetry breaking field in
the replica theory Bao  _et al._ (2020)). The addition of the antinoise terms
considered here can restore the symmetries of the model on average, allowing
for a stable ordered phase. Thus, our analysis indicates that MIPT for $D\geq
2$ can be made fault-tolerant via error mitigation techniques.

More broadly, the existence of error mitigation thresholds has important
implications for near-term quantum simulation and quantum computing. A natural
question is whether the threshold behavior observed here extends to other
models. For example, in quantum simulation of Hamiltonian dynamics, one can
apply a continuous time version of PEC Sun  _et al._ (2021). An error
mitigation threshold qualitatively similar to the one studied in this work
should arise for chaotic Hamiltonian evolution with sufficiently strong
interactions relative to the dissipation, raising the intriguing possibility
of a rich interplay between the Hamiltonian and error mitigated dynamics.
Similarly, error mitigation thresholds may naturally arise in more
conventional quantum algorithms like Shor’s factoring algorithm.

###### Acknowledgements.

We thank Chris Baldwin, Gregory Bentsen, Abhinav Deshpande, Bill Fefferman,
Alexey Gorshkov, David Huse, Zlatko Minev, Alireza Seif, Sagar Vijay, and
Brayden Ware for helpful discussions. Work supported in part by NSF QLCI grant
OMA-2120757, by NSF DMR-1653271, and NSF grant PHY-1748958.

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##  Appendix A Mean-Field Theory in Brownian Circuits

Here, we develop a mean-field theory for the error mitigation threshold in the
noisy-mitigated Brownian circuit model.

The unitary dynamics in the Brownian quantum circuit model is described by a
stochastic Hamiltonian

$H(t)=\sum_{i<j,\mu,\nu}J_{ij\mu\nu}(t)\sigma_{i}^{\mu}\sigma_{j}^{\nu},$ (9)

where $\sigma_{i}^{\mu}$ are Pauli operators $\mu\in\\{X,Y,Z\\}$ for site $i$
of $N$ qubits and $J_{ij\mu\nu}(t)$ is a white-noise correlated coupling with
variance Lashkari  _et al._ (2013); Jian  _et al._ (2022)

$\langle
J_{ij\mu\nu}(t)J_{k\ell\gamma\delta}(t^{\prime})\rangle=\frac{J}{2N}\delta_{ik}\delta_{j\ell}\delta_{\mu\gamma}\delta_{\nu\delta}\delta(t-t^{\prime}).$
(10)

The noise and antinoise are treated using a Lindblad master equation

$\dot{\rho}=-i[H(t),\rho]+\sum_{i}\frac{\gamma_{i}-\gamma_{a}}{4}\
\Big{(}-3\rho+\sum_{\mu}\sigma_{i}^{\mu}\rho\sigma_{i}^{\mu}\Big{)},$ (11)

where $\gamma_{i}$ is the local random noise rate and $\gamma_{a}$ is the
antinoise rate.

To analyze the dynamics we derive an effective master equation that describes
the replicated system

$M_{k}(\rho)=\mathbb{E}[U^{\otimes k}\rho U^{{\dagger}\otimes k}],$ (12)

where $U=\mathcal{T}e^{-i\int_{0}^{t}dt^{\prime}H(t^{\prime})}$ is the time
evolution operator under the Hamiltonian. Expanding to second order in an
infinitesimal time-step, we arrive at the equation
$M_{k}(\rho)=e^{\mathcal{L}_{k}t}$ for a Lindbladian $\mathcal{L}_{k}$ given
by

$\mathcal{L}_{k}(\rho)=\frac{J}{2N}\sum_{i\neq
j,\mu,\nu,r,s}\sigma_{ir}^{\mu}\sigma_{jr}^{\nu}\rho\sigma_{is}^{\mu}\sigma_{js}^{\nu}-\frac{1}{2}\\{\sigma_{ir}^{\mu}\sigma_{is}^{\mu}\sigma_{jr}^{\nu}\sigma_{js}^{\nu},\rho\\},$
(13)

where $r$ and $s$ are replica indices that run over $1,\ldots,k$. Thus, we
arrive at a master equation describing the average dynamics of the replicated
density matrices

$\dot{\rho}=\mathcal{L}_{k}(\rho)+\sum_{i}\frac{\gamma_{i}-\gamma_{a}}{4}\
\Big{(}-3k\rho+\sum_{r,\mu}\sigma_{ir}^{\mu}\rho\sigma_{ir}^{\mu}\Big{)}.$
(14)

To develop the mean field theory, we study the two-replica problem $k=2$.
Similar to a Haar random circuit, the Lindbladian $\mathcal{L}_{2}$ has two
steady states $I^{\otimes N}$ and $S^{\otimes N}$. As our mean-field ansatz,
we therefore use a product state of the form
$\rho=\bigotimes_{i=1}^{N}\rho_{i}$. A further simplification arises from the
nature of the dynamics that has an effective SU(2) symmetry in the average-
replica dynamics. As a result, we can express

$\rho_{i}=(1/4+\delta_{i})|s\rangle\langle s|+(1/4-\delta_{i}/3)P_{T},$ (15)

where $\delta_{i}$ is the deviation from an infinite temperature state,
$|s\rangle$ is a two-qubit singlet state across the two replicas, and $P_{T}$
is the projector onto the two-qubit triplet subspace of the two replicas.

The mean-field equations for $\delta_{i}$ take the form

$\dot{\delta}_{i}=-4\Big{[}\Delta_{i}+\frac{J}{N}\sum_{j\neq
i}(3+4\delta_{j})\Big{]}\delta_{i},$ (16)

where $\Delta_{i}=\gamma_{i}-\gamma_{a}$. This equation has the two steady-
state solutions $\delta_{i}=0$ and $\delta_{i}=-3/4$, corresponding to the
$I^{\otimes N}$ and $S^{\otimes N}$ solutions, respectively. In the case of
binary disorder, we define two populations of sites $A_{1/2}$ such that
$\Delta_{i}=\gamma_{1/2}-\gamma_{a}$, respectively, for $i\in A_{1/2}$. We
have the zero-mean field condition $p\Delta_{1}+(1-p)\Delta_{2}=0$. Defining
$G_{\pm}=\frac{1}{N}\Big{(}\sum_{i\in A_{2}}\delta_{i}\pm\sum_{i\in
A_{1}}\delta_{i}\Big{)}$, we arrive at simple closed set of equations in the
large-$N$ limit

$\displaystyle\dot{G}_{+}$
$\displaystyle=-4J(3+4G_{+})G_{+}+\frac{4|\Delta_{1}|}{2(1-p)}[G_{-}-(2p-1)G_{+}],$
$\displaystyle\dot{G}_{-}$
$\displaystyle=-4J(3+4G_{+})G_{-}+\frac{4|\Delta_{1}|}{2(1-p)}[G_{+}-(2p-1)G_{-}].$

Setting $p=1/2$, these equations reduce to the particularly simple form

$\displaystyle\dot{G}_{+}$
$\displaystyle=-4J(3+4G_{+})G_{+}+4|\Delta_{1}|G_{-},$ (17)
$\displaystyle\dot{G}_{-}$
$\displaystyle=-4J(3+4G_{+})G_{-}+4|\Delta_{1}|G_{+},$ (18)

For general $p$, the steady state solutions are
$G_{+}=0,-(3-|\Delta_{1}|/J)/4,-(3+\Delta_{2}/J)/4$. The all identity solution
$G_{+}=G_{-}=0$ only becomes an unstable fixed point for $|\Delta_{1}|/J\geq
3$, whereas it remains stable for weaker disorder. As a result, the phase
transition in the mean-field theory occurs at the disorder strength
$|\Delta_{1}|=3J$. Beyond this value of disorder, the mean-field steady-state
solution flows to an unphysical state; however, for weaker disorder, the
physical mean-field solution remains stable.

##  Appendix B Statistical Mechanics Mapping Formalism

Here, we review the statistical mechanics for the model with noise and
antinoise, generalizing the mappings studied in Ref. Nahum  _et al._ (2018);
Bao  _et al._ (2020); Jian  _et al._ (2020); Dalzell  _et al._ (2021, 2022).
In this paper, we focus our attention to calculating second-moment quantities
of a quantum state, which includes measures like fidelity, collision
probability, and linear cross entropy. The circuit-averaged calculations of
such quantities lends itself to a statistical mechanical mapping to an Ising
spin model.

Consider a second moment measure $M$, averaged over circuits from ensemble
$\mathcal{U}$. For a state a state $\rho_{C}$ of dimension $2^{2n}\times
2^{2n}$, generated using a circuit $C$ from an ensemble $\mathcal{U}$, a
circuit-averaged second moment measure $M$ for can be written as a two-copy
expectation of a linear operator $O_{M}$ of dimension $2^{4n}\times 2^{4n}$.

$\mathbb{E}_{C\in\mathcal{U}}[M[\rho_{C}]]=\mathbb{E}_{C\in\mathcal{U}}\left[\operatorname{tr}\left(O_{M}\rho_{C}\otimes\rho_{C}\right)\right]\\\
=\operatorname{tr}\left(O_{M}\mathbb{E}_{C\in\mathcal{U}}\left[\rho_{C}\otimes\rho_{C}\right]\right)$
(19)

The circuit-averaged two-copy state
$\mathbb{E}_{C\in\mathcal{U}}[\rho_{C}\otimes\rho_{C}]$, therefore, enables us
to calculated second-moment measures.

We consider circuit models that can be decomposed into a series of elementary
two-qubit gates (noisy or noiseless), each drawn independently from the two-
qubit Haar ensemble $\mathcal{U}_{2}$. The action of the circuit map on two-
copies of an initial input state is, thus, given by

$\rho_{C}{\otimes
2}={C}_{s}\circ{C}_{s-1}\circ\cdots\circ{C}_{1}[\rho_{0}\otimes\rho_{0}],$
(20)

where the elementary single-qubit or two-qubit channels are indexed using
integers $[1,s]$. For noiseless gates,
${C}_{i}[\rho_{0}\otimes\rho_{0}]=(C_{i}\otimes
C_{i})\rho(C_{i}^{\dagger}\otimes C_{i}^{\dagger})$. We can model a noisy
circuit by adding error channel $\mathcal{E}$ after each noiseless gate:

$\rho_{C}^{\otimes
2}=\mathcal{E}_{s}\circ{C}_{s-1}\circ\mathcal{E}_{s-1}{C}_{s-1}\circ\cdots\circ\mathcal{E}_{1}\circ{C}_{1}[\rho_{0}\otimes\rho_{0}].$
(21)

In general, the error channel may act on any set of qubits. For our purposes,
we assume that the error channel $\mathcal{E}_{i}$ acts on the qubit or the
pair of qubits acted on by the noiseless gate $C_{i}$. Since we draw each gate
from the random one-qubit or two-qubit Haar ensemble, we can replace the
noiseless maps $C_{i}[\rho]$ with the gate-averaged map
$\overline{C}_{i}[\sigma]=\mathbb{E}_{C\in\mathcal{U}_{1/2}}C[\sigma]$,

$\mathbb{E}_{C\in\mathcal{U}}[\rho_{C}\otimes\rho_{C}]=\mathcal{E}_{s}\circ\overline{C}_{s-1}\circ\mathcal{E}_{s-1}\overline{C}_{s-1}\circ\cdots\circ\mathcal{E}_{1}\circ\overline{C}_{1}[\rho_{0}\otimes\rho_{0}].$
(22)

The action of a random single-qubit gate $C$, on a state residing in the two-
copy Hilbert space is given by

$\overline{C}^{(1)}[\sigma]=\frac{\operatorname{tr}((1-S/2)\sigma)}{3}I+\frac{\operatorname{tr}((S-1/2)\sigma)}{3}S,$
(23)

where $I$ and $S$ are the $4\times 4$ identity matrix and SWAP matrices,
respectively. Similarly, the action of a random two-qubit gate on two copies
of a qubit-pair is given by

$\overline{C}^{(2)}[\sigma]=\frac{\operatorname{tr}((1-SS/4)\sigma)}{15}II+\frac{\operatorname{tr}((SS-1/4)\sigma)}{15}SS,$
(24)

where we use the shorthand $SS=S\otimes S$ and $II=I\otimes I$. If two copies
of a quantum state can be represented by a string of $\rho^{\otimes
2}\in\\{I,S\\}^{n}$, a single-qubit gate acts on qubit $k$ by modifying the
$j$th bit of the I-S string using the transition rules

$I\to I\qquad S\to S,$ (25)

while leaving the rest of the bits in the string unchanged. Similarly, a two-
qubit gate acting on qubits $j$ and $k$ modifies the $j$th and $k$th bits
according to the transition rules:

$II\to II\qquad IS,SI\to\frac{2}{5}(II+SS)\qquad{S}{S}.\to{S}{S}.$ (26)

A noiseless random circuit, therefore, can be represented as a linear operator
acting on the reduced space spanned by basis elements in $\\{I,S\\}^{n}$, with
each noiseless single-qubit gate given an identity map, and a two-qubit gate
given by the transition matrix

$T[\overline{C}^{(2)}]=\begin{pmatrix}1&2/5&2/5&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&2/5&2/5&1\end{pmatrix}.$ (27)

We can similarly, find the transition matrices corresponding to the noise and
antinoise channel. The single-qubit depolarization channel with error rate
$q$, given by the following map

$\mathcal{E}^{(1)}(\rho)=(1-q)\rho+q\operatorname{tr}(\rho)\frac{\mathbb{1}}{2},$
(28)

acts on two copies of a qubit such that

$\mathcal{E}[I]=\mathcal{E}[I_{2}\otimes
I_{2}]=\mathcal{E}^{(1)}[I_{2}]\otimes\mathcal{E}^{(1)}[I_{2}]=I_{2}\otimes
I_{2}=I$ (29)

, where $I_{2}$ is a $2\times 2$ identity matrix. Similarly,

$\displaystyle\mathcal{E}$ $\displaystyle[S]$
$\displaystyle=(\mathcal{E}^{(1)}\otimes\mathcal{E})[I_{2}\otimes
I_{2}+X\otimes X+Y\otimes Y+Z\otimes Z]/2$ $\displaystyle=\left[I_{2}\otimes
I_{2}+(1-q)^{2}\left(X\otimes X+Y\otimes Y+Z\otimes Z\right)\right]/2$
$\displaystyle=\left[(1-(1-q)^{2})/2I+(1-q)^{2}S\right]$

where we have used the fact that $S=(I_{2}\otimes I_{2}+X\otimes X+Y\otimes
Y+Z\otimes Z)/2$. The transition matrix corresponding to a depolarizing noise,
in the statistical mechanical picture is given by,

$T[\mathcal{E}_{q}]=\begin{pmatrix}1&(1-(1-q)^{2})/2\\\
0&(1-q)^{2}\end{pmatrix}$ (30)

Likewise, the antinoise channel of strength $q_{a}$, given by

$\mathcal{A}^{(1)}(\rho)=\frac{1}{1-q_{a}}\left(\rho-
q_{a}\operatorname{tr}(\rho)\frac{\mathbb{1}}{2}\right),$ (31)

acts on two-copies of a qubit such that

$\displaystyle\mathcal{A}[I]$
$\displaystyle=(\mathcal{A}^{(1)}\otimes\mathcal{A}^{(1)})[I_{2}\otimes
I_{2}]=I,\text{ and}$ (32) $\displaystyle\mathcal{A}[S]$
$\displaystyle=\left[\left(1-\frac{1}{(1-q_{a})^{2}}\right)\frac{I}{2}+\frac{1}{(1-q_{a})^{2}}S\right],$
(33)

giving a transition matrix

$T[\mathcal{A}_{q_{a}}]=\begin{pmatrix}1&\left(1-(1-q_{a})^{-2}\right)/2\\\
0&{(1-q_{a})^{-2}}.\end{pmatrix}$ (34)

Concatenating the transition matrix for the noise channel and the antinoise
channel gives the transition matrix for the composite channel.
$T[\mathcal{A}_{q_{a}}\circ\mathcal{E}_{q}]=T[\mathcal{A}_{q_{a}}].T[\mathcal{E}_{q}]$.

In our simulations, we start with an initial state drawn from the Haar
ensemble or a random product state. A Haar random state is proportional to
$I^{\otimes n}+S^{\otimes n}$ in the two-copy descrpition, while a random
product state is proportional to $(I+S)^{\otimes n}$. We can then use the
statistical mechanical formalism discussed above to evolve this state using
the respective transition matrices for two-qubit gates, noise and antinoise
channels.

##  Appendix C Local Probe

In Fig. 2, we presented different correlation metrics between two qubits in
the system as a probe of the phase transition. Here, we present an alternative
metric based on the entropy of a single qubit in the system.

Deep in the below threshold phase, this single-qubit entropy quantity
saturates to one bit, while above threshold it diverges to large negative
values due to the unphysical density matrix. As a result, we expect a crossing
to occur at the phase transition. Numerical simulations of the two-replica
stat-mech model for the all-to-all circuit model illustrate this behavior. In
Fig. 3(a), we show the unscaled behavior of the entropy for different system
sizes, which shows a crossing near $\sigma_{c}/\bar{q}=0.65(5)$. Collapsing
the data with this value of $\sigma_{c}$ fixed, we estimate a critical
exponent $\mu=1.0(2)$. In Fig. 2(c-d), we fixed $\mu=1$ in collapsing the
mutual information data based on these scaling results. This single-qubit
quantity also has advantages for experimental probes of the transition as it
requires minimal tomographic overhead to estimate.

Figure 3: Local probe of the error mitigation threshold: (a) Entanglement
entropy of a single-site in the system for different system sizes. The larger
sizes begin to develop a crossing, indicative of the phase transition. (b)
Scaling collapse for $\sigma_{c}/\bar{q}=0.65(5)$ and $\mu=1.0(2).$

##  Appendix D Error Mitigated Fidelity Benchmarks

Here, we introduce mitigated fidelity benchmarks and demonstrate an
exponential improvement in a mitigated verison of the linear cross-entropy
benchmark below the error mitigation threshold.

The task of sampling from the output distribution of a noiseless random
circuit is widely conjectured to be intractable with classical computers Boixo
_et al._ (2018); Bouland  _et al._ (2019); Movassagh (2018); Hangleiter and
Eisert (2022). This conjecture forms the basis for claims of achievement of
quantum computational advantage in recent experiments Arute  _et al._ (2019);
Wu  _et al._ (2021); Zhu  _et al._ (2021). The experimental claims remain
controversial, however, partly because of the effects of noise on the output
that render the signal classically simulatable at high depth Gao and Duan
(2018); Bouland  _et al._ (2021); Aharonov  _et al._ (2022). To provide
evidence that the output signal still remains close to the ideal case, one can
estimate fidelity benchmarks using the samples from the experiment. Verifying
the claim of computational advantage in the case of noisy circuits then
reduces to the task of achieving a sufficiently high “score” on the benchmark
Aaronson and Gunn (2019).

Recall that the linear XEB is given by the formula

$F_{\rm XEB}=2^{n}\sum_{x}p_{n}(x)p_{0}(x)-1,$ (35)

where $p_{0}(x)=|\langle x|U_{d}\cdots U_{1}|0\rangle|^{2}$ is the probability
of measuring outcome $x$ for the noiseless circuit of depth $d$ and
$p_{n}(x)=\langle
x|\mathcal{E}_{d}\circ\mathcal{U}_{d}\circ\cdots\mathcal{E}_{1}\circ\mathcal{U}_{1}(|0\rangle\langle
0|)|x\rangle$ is the analogous probability for the noisy circuit.

Figure 4: Dynamics of fidelity benchmarks above and below threshold: (a)
Disorder-averaged mean value of the logarithm of the circuit averaged
mitigated fidelity $-\langle\log\bar{F}_{M}\rangle/N$ and cross-entropy
benchmarking mitigated fidelity $-\langle\log F_{{\rm XEB}_{M}}\rangle/N$
above and below the error mitigation threshold in the all-to-all model. (b)
Dynamics of the standard deviation over the noise of the log-circuit-averaged
mitigated fidelities below threshold showing the improved scaling of the
typical log-fidelity as $\pm\mathcal{O}(\sqrt{Nd})$.

We now introduce the mitigated linear XEB, which is instead given by the
formula

$F_{\rm XEB_{M}}=2^{n}\sum_{x}p_{n}(x)p_{a}(x)-1,$ (36)

where $p_{a}(x)=\langle
x|\mathcal{A}\circ\mathcal{U}_{d}\circ\cdots\mathcal{A}\circ\mathcal{U}_{1}(|0\rangle\langle
0|)|x\rangle$ is a quasi-probability for the circuit with antinoise inserted
in place of noise. The quantity $p_{a}(x)$ can be computed on a classical
computer, which leads to a sampling formula for $F_{\rm XEB_{M}}$ using $M$
samples $x_{i}$ obtained from $p_{n}(x)$

$F_{\rm XEB_{M}}=\frac{2^{n}}{M}\sum_{i=1}^{M}p_{a}(x_{i})-1.$ (37)

This formula illustrates that the mitigated fidelity can be obtained without
directly implementing PEC except in purely classical post-processing. After
circuit averaging, one can quickly show that for depolarizing noise and its
antinoise partner, the mitigated fidelity is equivalent to the formula

$\bar{F}_{{\rm XEB}_{M}}=2^{n}\sum_{x}p_{0}(x)p_{an}(x)-1,$ (38)

where $p_{an}(x)=\langle
x|\mathcal{A}\circ\mathcal{E}_{d}\circ\mathcal{U}_{d}\circ\cdots\mathcal{A}\circ\mathcal{E}_{1}\circ\mathcal{U}_{1}(|0\rangle\langle
0|)|x\rangle$ implements the antinoise on the same copy as the noise. This
identity follows because the noise and antinoise on one copy have the
identical effect on the $I$ and $S$ operators after averaging over circuits.
From this expression, we see that, in the case of perfect mitigation,
$\bar{F}_{{\rm XEB}_{M}}$ reduces to its ideal value.

We can also define a mitigated fidelity that takes the form

$\displaystyle F_{M}$
$\displaystyle=\operatorname{tr}[\mathcal{A}\circ\mathcal{U}_{d}\circ\cdots\circ\mathcal{A}\circ\mathcal{U}_{1}(|0\rangle\langle
0|)$
$\displaystyle\times\mathcal{E}_{d}\circ\mathcal{U}_{d}\circ\cdots\circ\mathcal{E}_{1}\circ\mathcal{U}_{1}(|0\rangle\langle
0|)].$

In the case where $\mathcal{E}_{i}=\mathcal{A}^{-1}$ for every $i$, we can see
that $F_{M}=1$.

In Fig. 4], we show that the log-fidelity at low noise rates and $d$ grows as
$\mathcal{O}(Nd)$, whereas using error mitigation this scaling can be improved
to $\mathcal{O}(\sqrt{Nd})$, representing an exponential improvement in the
score that brings it closer to the $\mathcal{O}(d)$ scaling of the log-total
variation distance Deshpande  _et al._ (2022). Moreover, as explained above,
the cross-entropy benchmark fidelity can be mitigated entirely in classical
post-processing. As a result, the mitigated XEB fidelity can be estimated with
existing experimental data from random circuit sampling experiments.


# A misleading naming convention: de Sitter ‘tachyonic’ scalar fields

Jean-Pierre<EMAIL_ADDRESS>Hamed<EMAIL_ADDRESS>1Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris,
France  2Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences, Acad. G. Bonchev Str. Bl. 8, 1113, Sofia, Bulgaria

###### Abstract

We revisit the so-called ‘tachyonic’ scalar fields in de Sitter (dS)
spacetime. Through a rigorous group-theoretical analysis, we argue that
labeling these fields as ‘tachyonic’ lacks physical relevance, frequently
resulting in misinterpretations in scholarly discourse, such as the
misconception of faster-than-light travel associated with the corresponding
particles.

Let us begin by recalling the most comprehensive action on the dS background,
resulting in a linear equation of motion for a dS scalar field $\varphi$:

$\displaystyle
S(\varphi)=\frac{1}{2}\int_{\text{dS}}\left(g^{\mu\nu}\partial_{\mu}\varphi\,\partial_{\nu}\varphi-m^{2}_{H}\varphi^{2}\right)\mathrm{d}\Omega\,,$
(1)

where $g_{\mu\nu}$, with $\mu,\nu=0,1,2,3$, stands for the dS metric,
$\mathrm{d}\Omega$ for the invariant measure on dS, $R=12H^{2}$ for the Ricci
(or curvature) scalar, $m_{H}$ for the ‘mass’ term. Note that the latter
parameter, strictly speaking, $m^{2}_{H}$, encompasses the coupling term to
the curvature, which is conventionally denoted in literature by $\xi R$, where
$\xi$ denotes the coupling constant. The equation of motion derived from this
action reads as:

$\displaystyle\left({\Box_{H}}+m^{2}_{H}\right)\varphi=0\,.$ (2)

By definition (see, for instance, Refs. Tach1 ; Tach2 ), the ‘tachyonic’
scalar fields in $(1+3)$-dimensional dS spacetime are characterized by their
squared ‘masses’ being negative and assuming an infinite set of discrete
values, as outlined below:

$\displaystyle m^{2}_{H}=-H^{2}n(n+3)\,,$ (3)

where $n=0,1,2,\ldots\;$. Note that the so-called dS ‘massless’ minimally
coupled scalar field stands as a particular example of such fields,
distinguished by $n=0$, and hence, $m_{H}=0$. As already pointed out, the
commonly accepted interpretation for the emergence of the real negative
squared ‘masses’ attributed to the dS ‘tachyonic’ fields is a would-be
particle traveling faster than light (see, for instance, Refs. Tach1 ; Tach2 ;
1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ).

This manuscript raises a critical question regarding the interpretation
outlined above: does the quantity $m_{H}$, as defined in Eq. (3), truly
signify a meaningful mass within the framework of dS relativity? Clearly, if
it does not, any conclusions drawn from this parameter would lack physical
relevance. The remainder of this manuscript endeavors to address this very
question, while maintaining simplicity by adhering to units where $c=1=\hbar$,
representing the speed of light and the Planck constant, respectively.

To begin, we highlight two well-established facts regarding the modern
formulation of elementary particles:

  * •

In Minkowski quantum field theory (QFT), the concept of (‘free’) particles
emerges as a consequence of Wigner’s classification of the unitary irreducible
representations (UIRs) of the Poincaré group, that is, the group of motions in
flat Minkowski spacetime Wigner1 ; Wigner2 . In this context, the rest mass
$m$ and the spin $s$ of an (Einsteinian) elementary particle stand as the two
invariants that characterize the corresponding UIR of the Poincaré group.

  * •

Introducing a specific curvature to spacetimes is the sole method for
deforming the Poincaré group. This deformation gives rise to the dS and anti-
dS relativity groups of motion, as elaborated in Refs. Bacry ; Levy . This
distinctive situation confers a unique status upon dS and anti-dS spacetimes,
establishing them as the singular family of curved spacetimes where, to a
certain extent, the extension of the particle concept aligns with Wigner’s
framework (see Refs. dSBook ; AdS , and references therein); UIRs of the dS
and anti-dS groups, similar to their common Poincaré (null-curvature)
contraction limit, are generally distinguished by two invariant parameters of
the spin and energy scales.

Note that the latter fact reminds us of the Fronsdal principle (1965) Fronsdal
: “ _A physical theory that treats spacetime as Minkowskian flat must be
obtainable as a well-defined limit of a more general physical theory, for
which the assumption of flatness is not essential._ ”

With this foundation, let us delve deeper into dS relativity. The
$(1+3)$-dimensional dS spacetime can be effectively represented as a
hyperboloid situated within a $(1+4)$-dimensional ambient Minkowski spacetime,
denoted as $\mathbb{R}^{5}$:

$\displaystyle\text{dS}=\left\\{x\in\mathbb{R}^{5}\;;\;(x)^{2}\equiv\eta_{\alpha\beta}x^{\alpha}x^{\beta}=-H^{-2}\right\\}\,,$
(4)

where $\eta_{\alpha\beta}=\mbox{diag}(1,-1,-1,-1,-1)$, with
${\alpha},{\beta}=0,1,2,3,4$, and $H$ refers to the Hubble constant. The
latter determines the rate of expansion for the spatial sections of dS. In
this context, the dS metric is obtained by inducing the natural metric of
$\mathbb{R}^{5}$ onto the dS hyperboloid:

$\displaystyle\mathrm{d}s^{2}=\eta_{\alpha\beta}\mathrm{d}x^{\alpha}\mathrm{d}x^{\beta}\Big{|}_{(x)^{2}=-H^{-2}}=g_{\mu\nu}\mathrm{d}X^{\mu}\mathrm{d}X^{\nu}\,,$
(5)

where $X^{\mu}$ ($\mu,\nu=0,1,2,3$) denotes the four local spacetime
coordinates of the dS hyperboloid.

The dS relativity group is SO${}_{0}(1,4)$ (representing the connected
subgroup of the identity in O$(1,4)$) or any of its covering groups. The Lie
algebra corresponding to this group is manifested through the linear span of
the (ten) Killing vectors:

$\displaystyle
K_{\alpha\beta}=x_{\alpha}\partial_{\beta}-x_{\beta}\partial_{\alpha}\,.$ (6)

On the level of representation, these Killing vectors are represented by
(essentially) self-adjoint operators within the Hilbert space of functions
that are (spinor-)tensor-valued, and square-integrable under some invariant
inner product of Klein-Gordon type (or similar), defined on the dS spacetime
manifold:

$\displaystyle
K_{\alpha\beta}\;\mapsto\;L_{\alpha\beta}=M_{\alpha\beta}+S_{\alpha\beta}\,,$
(7)

where the orbital component is expressed as
$M_{\alpha\beta}=-\mathrm{i}(x_{\alpha}\partial_{\beta}-x_{\beta}\partial_{\alpha})$,
while the spinorial component $S_{\alpha\beta}$ operates on the indices of the
specified (spinor-)tensor-valued functions in a specific permutational manner.
The generator representatives $L_{\alpha\beta}$ adhere to the commutation
rules specified by the expression:

$\displaystyle\big{[}L_{\alpha\beta},L_{\varsigma\delta}\big{]}=-\mathrm{i}\left(\eta_{{\alpha}\varsigma}{L_{{\beta}\delta}}+\eta_{{\beta}\delta}{L_{{\alpha}\varsigma}}-\eta_{{\alpha}\delta}{L_{{\beta}\varsigma}}-\eta_{{\beta}\varsigma}{L_{{\alpha}\delta}}\right)\,.$
(8)

Within this framework, two Casimir operators emerge:

$\displaystyle Q^{(1)}=-\frac{1}{2}L_{\alpha\beta}L^{\alpha\beta}\,,\quad
Q^{(2)}=-W_{\alpha}W^{\alpha}\,,$ (9)

where the
$W_{\alpha}=-\frac{1}{8}{\varepsilon}_{\alpha\beta\varsigma\delta\rho}L^{\beta\varsigma}L^{\delta\rho}$,
while ${\varepsilon}_{\alpha\beta\varsigma\delta\rho}$ referees to the five-
dimensional totally antisymmetric Levi-Civita symbol. The Casimir operators,
by construction, commute with all generator representatives $L_{\alpha\beta}$,
and hence, they remain constant across all states within a given dS UIR.
Consequently, the classification of dS UIRs can be efficiently accomplished
based on the respective eigenvalues of these Casimir operators Thomas ; Newton
; Takahashi ; Dixmier ; Martin .

To elaborate this classification, let us focus more technically on the
specific context of dS scalar UIRs (relevant to the current study), realized
in the Hilbert space of Klein-Gordon square-integrable functions $\varphi(x)$
defined on the dS spacetime manifold. In this context, the quartic Casimir
operator $Q^{(2)}_{0}$333Note that the subscript ‘0’ indicates that the
carrier space is composed of scalar functions. vanishes and the quartic one
explicitly reads as:

$\displaystyle
Q^{(1)}_{0}=-\frac{1}{2}M_{\alpha\beta}M^{\alpha\beta}=-H^{-2}\Box_{H}\,,$
(10)

where ${\Box_{H}}=\partial_{\alpha}\partial^{\alpha}$ is the d’Alembertian
operator. We symbolize its action on the states $\varphi(x)$ carrying a
specified dS scalar UIR as:

$\displaystyle
Q^{(1)}_{0}\varphi(x)\;\left(=-H^{-2}\Box_{H}\varphi(x)\right)=\langle
Q^{(1)}_{0}\rangle\varphi(x)\,,$ (11)

where $\langle Q^{(1)}_{0}\rangle$ represents the corresponding Casimir
eigenvalue. Accordingly, the classification of dS scalar UIRs can be
effectively carried out based on the corresponding Casimir eigenvalues
$\langle Q^{(1)}_{0}\rangle$ Dixmier ; Takahashi :

  * •

The principal series $U_{0,\nu}$, characterized by $\langle
Q^{(1)}_{0}\rangle=\frac{9}{4}+\nu^{2}$, where $\nu\in\mathbb{R}$. Note that,
for a given $\nu$, two representations $U_{0,\nu}$ and $U_{0,-\nu}$ are
equivalent. In the null-curvature limit, these representations contract to the
massive scalar UIRs of the Poincaré group, comprehensively covering the entire
set of the latter and earning the designation of dS massive scalar
representations Mickelsson ; Garidi .

  * •

The complementary series $V_{0,\nu}$, characterized by $\langle
Q^{(1)}_{0}\rangle=\frac{9}{4}-\nu^{2}$, where $\nu\in\mathbb{R}$ and
$0<|\nu|<\frac{3}{2}$. Akin to the principal case, for any given $\nu$ within
the allowed range, the representations $V_{0,\nu}$ and $V_{0,-\nu}$ are
equivalent. Note that $V_{0,\nu=\frac{1}{2}}$ denotes the massless scalar
representation of the dS group, characterized by its unique extension to the
conformal group scalar UIR Barut ; Mack . Notably, this extension perfectly
corresponds to the conformal extension of the massless scalar UIR of the
Poincaré group Barut ; Mack .

  * •

The discrete series $\Pi_{p,0}$, characterized by $\langle
Q^{(1)}_{0}\rangle=(p+2)(-p+1)$, where $p=1,2,\ldots\;$.

_Remark 1: All dS UIRs, not falling strictly into the massive or massless
categories, either possess a nonphysical Poincaré contraction limit or lack
such a limit altogether._

_Remark 2: Practically speaking, in this group-theoretical context, the
Hilbert space carrying a given dS scalar UIR is densely generated by all
Klein-Gordon square-integrable eigenfunctions of the eigenvalue equation (11),
specifically adapted to the respective Casimir eigenvalue. As a result, Eq.
(11) assumes a fundamental role as the ‘field (wave) equation’._

Now, let us consider what serves as a universal replacement for the notion of
mass within the framework of dS relativity. In addressing this significant
query, we turn our attention to the Garidi mass formula GaridiMass . This
formula, formulated in terms of the invariant parameters that characterize the
UIRs of the dS group, offers a consistent and precise definition of mass in dS
relativity. It enables us to meaningfully distinguish between dS massive and
massless fields by aligning them with their counterparts in flat Minkowski
spacetime. Additionally, it holds the advantage of comprehensively
encompassing all previously introduced mass formulas within the dS context
(for further elucidation, see Refs. dSBook ; GaridiMass ).

In technical terms, for a given spin-$s$ dS UIR/field, the Garidi mass formula
precisely represents the difference between the eigenvalue of the quadratic
Casimir operator associated with the UIR/field and the eigenvalue of the
quadratic Casimir operator for the corresponding massless spin-$s$ scenario
(up to the physical dimension). Let us delve deeper into this formula by
exploring the case of the dS scalar, which holds particular relevance to our
study:

$\displaystyle\mathfrak{m}^{2}_{H}=H^{2}\left(\langle
Q^{(1)}_{0}\rangle-\langle Q^{(1)}_{0}\rangle_{\text{massless}}\right)\,,$
(12)

where $\langle Q^{(1)}_{0}\rangle$ denotes the Casimir eigenvalue linked with
the scalar representation/field under scrutiny, and $\langle
Q^{(1)}_{0}\rangle_{\text{massless}}=2$ signifies the eigenvalue corresponding
to the massless scalar representation $V_{0,\nu=\frac{1}{2}}$.

The Garidi mass definition, exemplified by the scalar case (12), holds
significance across all strictly dS massless and massive UIRs/fields. By
construction, it trivially reduces to zero for all massless UIRs/fields, and
remains positive for any massive UIR/field, such that, in the null-curvature
limit, consistently converges to the Minkowski mass $m$ of the respective
scalar, positive energy, Poincaré massive UIR $\mathcal{P}^{>}_{0,m}$. The
latter point can be easily checked for the massive scalar case, by
substituting $\langle Q^{(1)}_{0}\rangle=\frac{9}{4}+\nu^{2}$
($\nu\in\mathbb{R}$) into the mass definition (12):

$\displaystyle\mathfrak{m}_{H}=H\left(\nu^{2}+\frac{1}{4}\right)^{\frac{1}{2}}=H|\nu|\left(1+\frac{1}{4\nu^{2}}\right)^{\frac{1}{2}}\,.$
(13)

Then, by defining the asymptotic relation between the dS-invariant parameter
$\nu$ and the Poincaré-Minkowski mass $m$ as $m\sim H|\nu|$ and letting
$H\rightarrow 0$ and $\nu\rightarrow\infty$, we derive the following under the
null-curvature limit:

$\displaystyle\mathfrak{m}_{H}\;\rightarrow\;m\,.$ (14)

Thus, at the limit, the Garidi mass $\mathfrak{m}_{H}$ arises as the proper
mass $m$ of the elementary particle under consideration.

As a matter of fact, if we accept Einstein’s assertion that the proper mass of
an elementary particle remains unaffected by the curvature of spacetime, we
should regard the Garidi mass $\mathfrak{m}_{H}$ as equal to the proper mass
$m$, regardless of the curvature. Essentially, there should be no discernible
distinction between inertial and gravitational mass. This requirement implies
that the asymptotic relationship mentioned above between the dS-invariant
parameter $\nu$ and the Poincaré-Minkowski mass $m$, expressed as $m\sim
H|\nu|$, should consistently be stated as:

$\displaystyle\mathfrak{m}_{H}=H|\nu|\left(1+\frac{1}{4\nu^{2}}\right)^{\frac{1}{2}}=m\,.$
(15)

See Ref. DMGazeau  for more detailed discussions.

Considering the context provided above, it is also tempting to briefly discuss
the status of the Garidi mass formula for those dS scalar UIRs/fields that do
not strictly fall into the massive or massless categories. In the case of the
complementary scalar UIRs/fields, excluding the strictly massless one
$V_{0,\nu=\frac{1}{2}}$ where $\mathfrak{m}_{H}=0$, the bounded nature of the
parameter $\nu$ labeling these UIRs/fields renders the definition of a
Poincaré contraction limit technically unfeasible. Similarly, the same holds
true for the discrete scalar UIRs/fields, as the labeling parameter $p$ for
each UIR/field remains fixed. This observation exemplifies another crucial
point in our reasoning, as we elucidate below.

_Remark 3: The Garidi mass definition is significant as a universal proper
mass only for dS UIRs/fields that hold meaning from the Minkowskian
perspective. This includes both the dS massive and massless UIRs/fields.
However, for those dS UIRs/fields lacking a meaningful Minkowskian
interpretation, one can still apply the Garidi mass formula, albeit without a
physical interpretation._

With all the preceding information in mind, we are now poised to address the
initial question raised concerning the ‘tachyonic’ interpretation of the
scalar fields delineated by the (squared) ‘mass’ term (3). A direct comparison
between the field equation (2), incorporating the (squared) ‘mass’ term (3),
and its counterpart (11) (the latter derived within the context of dS group
representations) promptly illustrates that the ‘tachyonic’ scalar fields align
with the discrete scalar UIRs $\Pi_{p,0}$, where $p=1,2,\ldots$, with the dS
‘massless’ minimally coupled scalar field corresponding to the ‘lowest limit’
of these representations $\Pi_{1,0}$. The Garidi (squared) mass for these
UIRs/fields takes the following negative values:

$\displaystyle\mathfrak{m}^{2}_{H}=H^{2}\big{(}(p+2)(-p+1)-2\big{)}\,,$ (16)

where $p=1,2,\ldots\;$. However, as already pointed out, apart from those dS
UIRs/fields labeled as strictly massive or massless, the Garidi mass for all
UIRs/fields does not represent a genuine physical mass in the sense given by
Wigner, as it is not associated (either through contraction or conformal
extension) with the (‘true’) mass invariant parameter that labels the Poincaré
scalar UIRs. Frankly speaking, in these cases, it serves just as a parameter
in the theory, devoid of further significance. Consequently, any physical
interpretation derived from the real negative squared of this parameter, such
as faster-than-light travel, is totally irrelevant. The same argument holds
for the naming convention ‘massless’ minimally coupled scalar field, as its
corresponding representation $\Pi_{1,0}$ deviates from the aforementioned
strictly massless category of the dS UIRs — it is not, in fact, a massless
field.

Here, it is crucial to underscore that the preceding statement does not imply
that dS UIRs/fields without a Minkowskian counterpart hold no physical
significance. On the contrary, it is entirely valid to explore all dS
UIRs/fields within a unified framework, considering both mathematical (group
representation) and physical (field quantization) perspectives, albeit without
referencing concepts, such as mass, whose very existence depends on Poincaré-
Minkowski relativity.

In the end, it is tempting to compare the two definitions of mass
$\mathfrak{m}_{H}$ and ${m}_{H}$, particularly in the context of the dS scalar
case. A straightforward evaluation of Eqs. (2), (11), and (12) unveils the
following relationship:

$\displaystyle\mathfrak{m}^{2}_{H}={m}^{2}_{H}-2H^{2}\,.$ (17)

Evidently, in the strictly massless scalar case $V_{0,\nu=\frac{1}{2}}$, where
$\mathfrak{m}_{H}=0$, the nonvanishing nature of the ‘mass’ parameter $m_{H}$
is misleading; notably, $m_{H}$ diminishes to zero in the scenario of the
‘massless’ minimally coupled scalar field!

## Akcnowledgment

Hamed Pejhan is supported by the Bulgarian Ministry of Education and Science,
Scientific Programme ‘Enhancing the Research Capacity in Mathematical Sciences
(PIKOM)’, No. DO1-67/05.05.2022.

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# Cosmological Constant and
Maximum of Entropy for de Sitter Space

Igor Volovich

###### Abstract

There are at least two cosmological constants calling for explanation. The
first one describes the quasi-de Sitter inflation in the early universe, and
the second describes the current acceleration of the universe associated with
dark energy. An approach to the computation of the inflationary cosmological
constant in the early universe is proposed. The tunneling and no-boundary
proposals suggest that the ground state of the early universe is the de Sitter
space. In this paper it is argued that the radius of the de Sitter space, i.e.
the cosmological constant, can be computed using the principle of maximum
entropy. The universe emerges from “nothing” that corresponds to a minimum of
entropy. The entropy reaches its maximal value for the 4-dimensional de Sitter
space with the inflationary cosmological constant
$\Lambda=3\pi\,\exp\\{-\psi(3/2)\\}$, where $\psi$ is the digamma function,
$\Lambda\approx 9.087$ in Planck units.

##  1 Introduction

Einstein introduced the cosmological constant [1] assuming that there is only
one cosmological constant, since there is only one universe. However now it is
assumed that there are at least two cosmological constants, the inflationary
cosmological constant describing the quasi-de Sitter phase in the early
universe and the second cosmological constant describing the current
acceleration of the universe associated with dark energy.

A possible connection between a cosmological constant and elementary particle
physics was considered by Zeldovich [2]. In particular, he suggested, based on
Sakharov’s previous work [3], a possible connection between the vacuum
fluctuations of quantum matter and the cosmological constant. The inflationary
cosmology is considered in [4, 5, 6].

In the present paper an approach to the computation of the inflationary
cosmological constant in the early universe is proposed. In quantum cosmology
[7, 8, 10, 9] the tunneling wave function describes the early universe and no-
boundary proposal corresponds to the ground state of the universe. In both
cases one has de Sitter space in semi-classical approximation. In this paper
it is argued that the radius of the de Sitter space, i.e. the cosmological
constant in the original action, can be computed if one uses the principle of
maximum entropy and also interprets the spacetime dimension as a thermodynamic
quantity.

In Sect.2, it will be shown that the universe, arising from “nothing” with
minimal entropy, reaches, for the 4-dimensional de Sitter space, the maximum
entropy value for the inflationary cosmological constant given by

$\Lambda=3\pi\,\exp\\{-\psi(3/2)\\},$ (1.1)

where $\psi$ the digamma function, $\Lambda\approx 9.087$ in Planck units. For
$0\leq D\leq 4$, entropy at $\Lambda$ given by (1.1) is a monotonic increasing
function of the number of dimensions, which can be said to play the role of
time.

The proposal that the area of the cosmological horizon should be interpreted
as an entropy was made in [11]. It is proposed that the entropy of the empty
de Sitter space maximises the de Sitter entropy [12, 13]. There exists a
maximum entropy state on de Sitter space. Time dependence of entropy of de
Sitter space is considered in [14, 15]. An algebra of observables of type
$II_{1}$ describing the maximum entropy state in the de Sitter space is given
in [16]. The correspondence between de Sitter space and Bose gases is
considered in [17]. Note that the expression for the entropy even for positive
$D$ includes the area of the unit sphere $S^{D-2}$, so the negative dimensions
are involved, see Appendix A and [17].

The paper is organized as follows. Cosmological constant in early universe is
computed in Sect.2. In conclusion initial data for inflation and the Planck
era are briefly discussed. In Appendix A remarks on complex and negative
dimensions are made.

##  2 Cosmological Constant

In this Section the inflationary cosmological constant $\Lambda$ for the early
universe is computed.

###  2.1 Preliminary

The de Sitter metric has the form

$ds^{2}=-dt^{2}+e^{2Ht}\left(dy_{1}^{2}+...+dy_{D-1}^{2}\right),$ (2.1)

here the Hubble parameter $H$ is related with the cosmological constant as

$\Lambda=\frac{1}{2}(D-1)(D-2)H^{2}.$ (2.2)

The metric on the static patch of de Sitter space is

$ds^{2}=-\left(1-\frac{r^{2}}{\ell^{2}}\right)dt^{2}+\left(1-\frac{r^{2}}{\ell^{2}}\right)^{-1}dr^{2}+r^{2}d\omega_{D-2}^{2}.$

here the de Sitter radius $\ell$ is $\ell=1/H$. The Hawking temperature and
the entropy are

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi\ell},$ (2.3)
$\displaystyle S$ $\displaystyle=$ $\displaystyle
S(l,D)=\frac{\ell^{D-2}}{4}\Omega_{D-2},$ (2.4)

here $\ell$ is the radius of $D$-dimensional de Sitter space and
$\Omega_{D-2}$ is the area of the sphere $S^{D-2}$

$\Omega_{D-2}=\frac{2\pi^{(D-1)/2}}{\Gamma(\frac{D-1}{2})}.$ (2.5)

###  2.2 Principle of maximal entropy

The principle of maximum entropy states that the probability distribution
which best represents the current state of knowledge about a system is the one
with largest entropy, see [18].

We will use the following modification of this principle which takes into
account the knowledge that the observed spacetime dimension is $D_{0}=4$. This
modified principle can be called the conditional principle of maximum entropy.
We formulate it as follows.

The de Sitter radius $\ell_{0}$ satisfies the condition that the maximum
entropy $S(\ell_{0},D)$ over the changing number of space-time dimensions is
realized at $D=4$ and this maximal value coincides with the entropy value
$S(\ell_{0},D_{0})$ for the observed spacetime dimension $D_{0}=4$, i.e.
$\ell_{0}$ such that

$\ell_{0}:\quad\max_{D}\,S(\ell_{0},D)=S(\ell_{0},D_{0}).$ (2.6)

###  2.3 Computation of $\Lambda$

The entropy of de Sitter space is given by

$S(\ell,D)=\frac{1}{4}\,\ell^{D-2}\Omega_{D-2}=\frac{1}{2}\,\ell^{D-2}\frac{\pi^{(D-1)/2}}{\Gamma((D-1)/2)}.$
(2.7)

We consider $D$ as a real variable. The derivative of entropy with respect to
$D$ is

$\displaystyle\frac{\partial S(\ell,D)}{\partial
D}=\frac{\pi^{\frac{D-1}{2}}\ell^{D-2}}{4\Gamma\left(\frac{D-1}{2}\right)}\left(\log(\pi\ell^{2})-\psi\left(\frac{D-1}{2}\right)\right),$
(2.8)

where $\psi$ is the digamma function, see eq.A.4.

For extremum of the entropy with respect to $D$ we have

$\frac{\partial S(\ell,D)}{\partial D}=0.$ (2.9)

Solving this equation for $D=4$ with respect to $\ell_{0}$ we find

$\ell_{0}^{2}=\frac{1}{\pi}\exp\\{\psi(3/2)\\}\approx 0.3301$ (2.10)

In Fig.1.A we plot the entropy as function of $D$ for different $\ell$. We see
that $S(\ell,D)$ for fixed $\ell$ and varying $D$ in the region $D>1$ define
positive functions with one maximum, which location depends on $\ell$.

The blue curve in Fig.1.A shows that for $\ell=1.0$ the maximum of the entropy
is at $D=8.257$, since $S(1,D)=\Omega_{D-2}/4$ and when $D$ varies, the
maximum area of unit spheres $S^{D}$ is reached at $D=6.257$, see Sect.A.1 and
Fig.3.

As we see from above formula, one can take $\ell=\ell_{0}$ such that for this
value of $\ell_{0}$ the maximum in $D$ is realized at $D=4$, see Fig.1.B. The
corresponding cosmological constant is

$\Lambda=\frac{3}{\ell_{0}^{2}}=3\pi\,\exp\\{-\psi(3/2)\\}\approx 9.087.$
(2.11)

For $D=1$ and all $\ell$ the entropy is equal to zero and for $0<D<1$ it
becomes negative for all $\ell$, see Fig.2.A.

A B

Figure 1: A)$S_{dS}$ as function of $D$ for different $\ell$: $\ell=1.2$ cyan
line, $\ell=1.0$ blue line and $\ell=0.8.$ red line. B) $S_{d}S$ as function
of $D$ for $\ell=\ell_{0}=0.575$. The vertical red line shows the location of
maximum entropy for $D=4$.

It is interesting to note that the entropy as function of $D$ and fixed $\ell$
given by equation (2.6) has also a minimum exactly at the zero dimension,
$D=0$. Indeed,

$[\log(\pi\ell^{2})-\psi\left(\frac{x-1}{2}\right)]\Big{|}_{\ell=\ell_{0},\,x=0}=0,$
(2.12)

see Fig.2.B. The value of entropy for $\ell=\ell_{0}$ and $D=0$ is negative

$S(\ell_{0},0)=-0.482084.$ (2.13)

For other $\ell$, the entropy values for $D=0$ are also negative, but the
minimum values of entropy are implemented for other values of $D$, see
Fig.2.A.

A B

Figure 2: A) $S_{dS}$ as function of $D$ for different $\ell$ near $D=0$:
$\ell=1.2$ the cyan line, $\ell=1.0$ the blue line, $\ell=0.8.$ the red line,
$\ell=\ell_{0}$ darker green line and $\ell=0.4$ the green line. B)
$S_{1}=\log(l^{2}\pi)-\psi\left(-\frac{1}{2}\right)$ as function of $\ell$. We
see that the line crosses the horizontal axis at the point
$\ell=\ell_{0}=0.575$.

###  2.4 Spacetime dimension and the first law of thermodynamics

We interpret the spacetime dimension $D$ as a thermodynamic quantity by
analogy with the number of particles $N$ and add the corresponding term to the
first law of thermodynamics, which now reads

$dF=-S\,dT+\nu d\,D.$ (2.14)

Here $\nu$ is a thermodynamics potential analogues to the chemical potential.

##  3 Conclusion

In this note the inflationary cosmological constant $\Lambda$ of early
universe is computed. It is given by eq.(2.11). For this computation the
principal of maximal entropy is used. The problem of initial conditions for
inflationary models [4, 5, 6] is considered in [19, 20]. One can use the
cosmological constant (2.11) as the initial condition for inflationary models.
In particular, for the Friedman equation for the Hubble parameter we mean the
following initial condition

$H(0)=\frac{1}{\ell_{0}},$ (3.1)

where $\ell_{0}$ is given by equation (2.10) can be used.

As a final remark let us note that here we discussed the creation of the
Universe from ”nothing” after the Planck era. On the Planck scale, the usual
consideration based on quantum gravity is not applicable, since there are
fluctuations in geometry, topology, and even number fields. For a discussion
of quantum cosmology related to the fluctuations of number fields during the
Planck era, see [21, 22] and references therein.

## Acknowledgements

I would like to thank I. Aref’eva, V. Belokurov, V. Berezin, B. Dragovich, E.
Dynin, V. Frolov, R. Kerner, Y. Kogan, M. Khramtsov, A. Linde, V. Zagrebnov
for fruitful discussions. This work is supported by the Russian Science
Foundation (project 19-11-00320, V.A. Steklov Mathematical Institute)

Appendix

##  Appendix A On Complex and Negative Dimensions

In this section examples of analytical continuation of dimensions of simple
geometrical objects and fundamental solutions of partial differential
equations are discussed.

###  A.1 Volumes and areas

The volume of the cube $C^{d}$ in ${\mathbb{R}}^{d}$ is
$\mathcal{V}(C^{d})=L^{d}$, where $L$ is the length of the side of the cube.
The area of the unit sphere $S^{d}$ is

$\Omega_{d}=\frac{2\pi^{(d+1)/2}}{\Gamma(\frac{d+1}{2})},$ (A.1)

where $\Gamma(z)$ is the gamma function. The functions $L^{d}$ and
$\Omega_{d}$ admit analytical continuations for all complex values of $d$, in
particular to negative $d$. In the Fig.3 the form of the function $\Omega_{d}$
is shown for real $d$.

A B

Figure 3: A) $\Omega_{d}$ as function of $d$. B) Zoom of A) for negative $d$.

For extremum of $\Omega_{d}$ with respect to $d$ we have

$\frac{\partial\Omega_{d}}{\partial d}=0,$ (A.2)

or explicitly,

$\frac{\pi^{\frac{d+1}{2}}\left(\log(\pi)-\psi\left(\frac{d+1}{2}\right)\right)}{\Gamma\left(\frac{d+1}{2}\right)}=0,$
(A.3)

where $\psi(z)$ is the digamma function,

$\psi(z)=\frac{d}{dz}\ln\Gamma(z).$ (A.4)

A solution $d=d_{0}$ of this equation for positive $d>0$ satisfies

$\log(\pi)-\psi\left(\frac{d_{0}+1}{2}\right)=0$ (A.5)

and $d_{0}=6.257$. At this point the $\Omega_{d}$ reaches the local maximum,
see Fig.3.A.

There are other solutions of (A.5) located at negative $d$,
$d=-1.77,-3.93,-6.01,...$, see Fig.3.B. Note that negative odd $d$, $d=-2n-1$,
do not provide the extremum of $\Omega_{n}$ since at these points infinities
of $\Gamma$ are compensated by infinities of $\psi$, see Fig.4.

Figure 4: $\Gamma(\frac{d+1}{2})$, $\psi(\frac{d+1}{2})$ and
$\psi(\frac{d+1}{2})/\Gamma(\frac{d+1}{2})$ as function of $d$.

The volume of the ellipsoid $E^{d}$:

$\sum_{i=1}^{d+1}\,\frac{x_{i}^{2}}{a_{i}^{2}}\leq 1$ (A.6)

is

$\mathcal{V}(E^{d})=\frac{\pi^{(d+1)/2}}{\Gamma(\frac{d+1}{2}+1)}\,a_{1}\cdot...\cdot
a_{d+1}.$ (A.7)

Take $a_{i}=e^{c^{i}}$, then

$\mathcal{V}(E^{d})=\frac{\pi^{(d+1)/2}}{\Gamma(\frac{d+1}{2}+1)}\,e^{\sum_{i=1}^{d+1}c^{i}}=\frac{\pi^{(d+1)/2}}{\Gamma(\frac{d+1}{2}+1)}\,e^{c\frac{1-c^{d+1}}{1-c}}.$

The volume $\mathcal{V}(E^{d})$ admits an analytical continuation for all
complex $d$ if $c>0$.

###  A.2 Fundamental solutions in negative dimensions

  * •

Fundamental solution of the Laplace equation in $\mathbb{R}^{d}$ has the form

$E_{d}(x)=\frac{r^{2-d}}{(2-d)\Omega_{d-1}},\quad\Delta E_{d}=\delta.$ (A.8)

It admits analytical continuation for complex $d$.

  * •

There is an analogous expression for fundamental solution (propagator) for
wave equation.

  * •

Fractional derivatives satisfy the following convolution relations

$\frac{x_{+}^{\lambda}}{\Gamma(\lambda+1)}*\frac{x_{+}^{\mu}}{\Gamma(\mu+1)}=\frac{x_{+}^{\lambda+\mu}}{\Gamma(\lambda+\mu+1)},$
(A.9)

which admit analytical continuation for complex $\lambda$ and $\mu$.

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