# How to engineer a quantum wavefunction

Peter W. Evans email<EMAIL_ADDRESS>School of Historical and Philosophical
Inquiry, University of Queensland Dominik Hangleiter email<EMAIL_ADDRESS>_Joint Center for Quantum Information and Computer Science (QuICS)_ ,
University of Maryland and NIST  Karim P. Y. Thébault email:
<EMAIL_ADDRESS>Department of Philosophy, University of Bristol

###### Abstract

In a conventional experiment, inductive inferences between source and target
systems are typically justified with reference to a uniformity principle
between systems of the same _material type_. In an analogue quantum
simulation, by contrast, scientists aim to learn about target quantum systems
of one material type via an experiment on a source quantum system of a
different material type. In this paper, we argue that such an inference can be
justified by reference to the two quantum systems being of the same _empirical
type_. We illustrate this novel experimental practice of _wavefunction
engineering_ with reference to the example of Bose-Hubbard systems.

###### Contents

  1. 1 Introduction
  2. 2 A Case Study of Analogue Quantum Simulation: Bose-Hubbard Physics
    1. 2.1 The Analogue Systems
      1. 2.1.1 Simulation Source System
      2. 2.1.2 Target System 1: Superfluid ^4He in Vycor
      3. 2.1.3 Target System 2: Triplons in Quantum Dimer Magnets
      4. 2.1.4 Target System 3: Cooper Pairs in Josephson Junction Arrays
    2. 2.2 Summary of Analogue Systems
    3. 2.3 Internal Validity of Analogue Quantum Simulations
  3. 3 Uniformity Principles in Analogue Quantum Simulation
    1. 3.1 Tokens, Types and External Validation
    2. 3.2 Material Types, Empirical Types, and Universality Types
    3. 3.3 Empirical Quantum Types
  4. 4 External validation of analogue quantum experiments
    1. 4.1 External validation of Bose-Hubbard analogue simulations
    2. 4.2 Universality-based external validation reconsidered
  5. 5 Discussion

##  1 Introduction

On what basis should we categorise quantum systems as tokens of the same type?
One option is to distinguish types of quantum systems by their material
constitution focusing on the specific details of the quantum state in
combination with state-independent properties, such as masses, charges, and
couplings, that designate the detailed physical properties of the system. Call
this the _material type_ view. The second option is to distinguish types of
quantum system by structural similarity in empirical behaviour. In particular,
we could take any two quantum systems to be of the same type when in some
specified parameter regime a set of experimental prescriptions result in
appropriately similar measurement outcomes. Call this the _empirical type_
view.

The particular relevance of the distinction between material type and
empirical type views arises in the context of analogue quantum simulations
wherein a ‘source’ quantum system is manipulated in the lab with the aim of
gaining understanding of a ‘target’ quantum system which is not directly
manipulated. Most significantly, the form of justification for the source-
target inferences involved in analogue quantum simulation is sensitive to how
wide we draw the category of types of quantum systems. When, as in the small
existent philosophical literature, something like the material type view is
assumed, we find that analogue quantum simulations by definition involve a
novel form of inter-type uniformity reasoning requiring justification by way
of universality arguments.111See, in particular, (Dardashti _et al._ , 2017,
2019; Thébault, 2019; Crowther _et al._ , 2019; Evans and Thébault, 2020; Gryb
_et al._ , 2021; Hangleiter _et al._ , 2022; Field, forthcoming). However, by
contrast, and as is often implicit in the physics literature, when something
like the empirical type view is assumed, a more conventional form of intra-
type uniformity reasoning is applied, albeit with an atypical notion of type.

In this paper we seek to characterise the implications of the empirical type
view within the scientific practice of analogue quantum simulation drawing
upon the specific example of Bose-Hubbard physics. We set out the
epistemological implications of interpreting such a practice as ‘wavefunction
engineering’, along the lines of the empirical type view. In particular, we
examine the justificatory arguments required with a focus on the role of
‘quantumness’ and robustness of phenomena within specific parameter regimes.
Such arguments justify simultaneous de-idealisation of a single abstract
quantum model to both source and target system models within a designated
range of applicability and in so doing provide justification for reasoning
based upon regularity within empirical types. We will conclude by considering
some attendant methodological concerns relating to the problem of cleanly
separating epistemological and ontological issues in the analysis of
scientific practice.

##  2 A Case Study of Analogue Quantum Simulation: Bose-Hubbard Physics

Analogue quantum simulation consists in the modelling of some ‘target’ quantum
system or model by some other well-controlled ‘source’ quantum system.
Successful analogue quantum simulation requires that the source and target
models are appropriate theoretical models between which a partial isomorphism
holds in the relevant regime of idealisation (Hangleiter _et al._ , 2022). The
key to the simulation is that the source system can be controlled more easily
than the target system, and so an experiment on the source system can probe
elements of the target system that are experimentally inaccessible, given that
the idealised models are appropriately verified.

One class of dynamical systems that are particularly ripe for modelling in
analogue quantum simulation experiments are those that conform to the Bose-
Hubbard model, which describes the ‘dynamics of a lattice’ of interacting
bosons. The discovery of quantum phase transitions at zero temperature between
a superconducting and an insulating phase in granular superconductors sparked
theoretical and experimental interest in the model (Bruder _et al._ , 2005, p.
566). This led to the experimental investigation of various other systems
described by the Bose-Hubbard Hamiltonian, including thin Helium-films and
arrays of Josephson junctions as well as more theoretical interest in the
model itself.

Remarkably, it was discovered that bosons loaded into an optical lattice can
be described by precisely the same model (Jaksch _et al._ , 1998). The
experimental accessibility of the bosonic system in an optical lattice allows
a greater range of experimental investigations of Bose-Hubbard dynamics to
take place. The potential of cold atoms as an analogue simulation platform was
then experimentally demonstrated by showing that they undergo the same phase
transition at zero temperature between a superfluid and an insulator phase
(Greiner _et al._ , 2002). The phase transitions in these very different
systems described by the Bose-Hubbard Hamiltonian are underpinned by the same
physical principles: that is, the ‘competition between the trend to global
coherence, due to the hopping of bosonic particles, and the tendency towards
localization induced by the strong interactions’ Bruder _et al._ (2005).

###  2.1 The Analogue Systems

####  2.1.1 Simulation Source System

The accessibility of the bosonic system in an optical lattice makes it an
ideal source system for analogue simulation. Such system are typically
constructed by employing counter-propagating lasers to form a space-dependent
lattice potential combined with a magneto-optical trap (MOT). The lattice
potential is used as a location grid in which ultra-cold atoms can be
positioned – for bosons, ^87Rb is a typical such atom – and the MOT is used to
confine the atomic cloud. In the right parameter regime, this system can be
described by the Bose-Hubbard Hamiltonian (Jaksch _et al._ , 1998):

$H_{\text{BH}}=-J\sum_{\langle
j,k\rangle}\left(b^{\dagger}_{j}b_{k}+b^{\dagger}_{k}b_{j}\right)+U\sum_{j}b^{\dagger}_{j}b^{\dagger}_{j}b_{j}b_{j}+\sum_{j}\mu_{j}b^{\dagger}_{j}b_{j}~{}.$

The Hamiltonian represents atoms by the bosonic creation and annihilation
operators $b^{\dagger}_{j}$ and $b_{j}$ at lattice site $j$, and has terms
representing the energy gain $J$ when atoms hop between neighbouring sites,
the energy cost $U$ of two atoms at the same site, and the energy offset
$\mu_{j}$ of each lattice site. Zero-temperature or ‘quantum’ phase
transitions can be understood as the transition between regimes in which one
of $J$ or $U$ dominates the model and, correspondingly, its ground state. When
$J$ dominates, hopping behaviour is much more likely to occur, and so the
ground state consists of delocalised bosons across the lattice. This is the
superfluid phase. When $U$ dominates, there is a strong local repulsion
between lattice sites that prevents global coherence. This is the Mott
insulator phase (Bruder _et al._ , 2005, p. 567).

What makes the cold-atom system suitable as a source system for the analogue
quantum simulation of the target systems we outline below is that it is
accessible to experimental manipulation and probing of a sort not possible for
the target systems. Not only can all the model parameters be tuned to produce
a variety of phenomena across the lattice potential, but location and momentum
information of the atoms in the lattice can be measured with remarkable
precision (Bruder _et al._ , 2005; Hangleiter _et al._ , 2022). The key
feature of these examples that we wish to emphasise is that there is a wide
variety of physical systems that are successfully targeted by this source
system.

####  2.1.2 Target System 1: Superfluid ^4He in Vycor

Vycor is a specially manufactured high-silica glass. When it is manufactured
as a porous structure, it is an ideal substrate for the study of confined
liquids in condensed matter physics. Helium-4 adsorbed in Vycor is observed to
form a superfluid: it behaves as an interacting ideal Bose gas that typically
results from the formation of a Bose-Einstein condensate (BEC) (Reppy, 1984).
Since the Bose-Hubbard model describes an interacting Bose gas in a lattice
that behaves as a superfluid below some critical temperature, one would expect
superfluid ^4He in Vycor to conform to the Bose-Hubbard model behaviour.

Indeed, at large ^4He densities a conventional phase transition between a
superfluid phase and a Mott insulator phase is observed at finite temperature
(Fisher _et al._ , 1989). The critical temperature, $T_{c}$, at which this
phase transition occurs decreases with the density $\rho$ of ^4He, reaching
$T_{c}=0$ at some positive density $\rho_{c}(T=0)$. At zero temperature, the
system then undergoes a transition from a Mott-insulating state to a
superfluid as the density $\rho$ crosses $\rho_{c}(T=0)$. This, and subsequent
(Weichman, 2008), observation of the quantum phase transition behaviour
constitutes empirical evidence that the Bose-Hubbard model with density-
dependent hopping and interaction parameters is a valid characterisation of
the system. As a result, this behaviour is structurally and formally similar
to the zero-temperature superfluid-insulator phase transition of the ^87Rb
atoms in the optical lattice.

####  2.1.3 Target System 2: Triplons in Quantum Dimer Magnets

Typical magnetic materials consist of an ordered arrangement of magnetic spin
states. For certain materials, so-called spin dimer compounds, pairs of spin
states couple and, due to the crystalline structure of the material, interact
only weakly with other coupled spin states. These weakly interacting ‘dimers’,
as they are known, generate a paramagnetic ground state in the material
comprised of local entangled spin singlet states, with an excitation gap to an
excited triplet state. When a high strength magnetic field is applied to the
material, Zeeman splitting of the triplet state closes the excitation gap, and
the entangled spin singlets transition to the excited triplet state, and the
material to a magnetically ordered state.

The dimers in such systems behave as ‘bosonic quasiparticles’ and, when
excited by a magnetic field, are known as ‘triplons’ (Nohadani _et al._ ,
2005, p. 1). Significantly, this phase transition from the paramagnetic phase
to the ordered phase can be described as the formation of a BEC. In the
appropriate parameter regime, the critical temperature of the transition
vanishes, and so this phase transition is analogous to the zero-temperature
transition from a Mott-insulating phase to a superfluid condensate. This
quantum phase transition behaviour has been unambiguously verified
experimentally (Rüegg _et al._ , 2003; Giamarchi _et al._ , 2008) and, as
such, provides empirical evidence that the Bose-Hubbard model is a valid
characterisation of the system.

####  2.1.4 Target System 3: Cooper Pairs in Josephson Junction Arrays

A Josephson junction array is an array of superconducting islands weakly
coupled by (Josephson) tunnel junctions. The superconducting behaviour of the
system is determined by the interplay between the strength of the coupling
energy between the islands and the strength of the electrostatic interaction
energy of Cooper pair charges at each island. When the (Josephson) coupling
energy as between the islands is high, the array system tends towards
superconducting coherence across the islands. On the other hand, a high
interaction energy of Cooper pairs as controlled by the island capacitance
favours charge localisation on each island, and so the array system tends
towards the suppression of superconducting coherence (Bruder _et al._ , 2005,
p. 569). The behaviour of Josephson tunnelling and the interaction of Cooper
pair charges is described by the quantum phase model Hamiltonian, which is
formally equivalent to the Bose-Hubbard Hamiltonian (Benatti _et al._ , 2008).

In the regime of high coupling energy, there is a critical temperature below
which the array system is in a globally coherent superconducting state – the
Cooper pairs ‘condense’ into the same ground state. However, in the regime
where the electrostatic interaction energy at each island is comparable to the
coupling energy between adjacent islands, lowering the temperature of the
array increases the resistance between islands, and the array undergoes a
transition to an insulator phase (even though each island is still
superconducting) (Bruder _et al._ , 2005, p. 569). This phase transition is
experimentally well explored (Geerligs _et al._ , 1989; van der Zant _et al._
, 1996), to the extent that the Bose-Hubbard model is taken as a valid
characterisation of the behaviour of the Josephson junction arrays, with the
Cooper pairs behaving as the bosons. As such, this phase transition is
analogous to a zero-temperature superfluid-insulator phase transition in the
optical lattice system.

###  2.2 Summary of Analogue Systems

All four of the systems discussed are well described by the Bose-Hubbard model
within an appropriate parameter regime. All four undergo an analogue zero-
temperature phase transition from an insulting phase to a
superfluid/superconductor phase. The source system consists of an array of
^87Rb in an optical lattice potential. The zero-temperature quantum phase
transition in this system is controlled by manipulation of the model
parameters $J$ or $U$, affected by varying an external magnetic field and the
amplitude and phase of the lasers generating the lattice potential (Hangleiter
_et al._ , 2022, p. 28). Target system 1 consists of ^4He adsorbed in porous
Vycor. The phase transition in this system is controlled by increasing the
density of ^4He at zero temperature. Target system 2 consists of entangled
spins states in a magnetic material – quantum dimers – whose transition to an
excited triplon state causes the material to transition from a paramagnetic
phase to an ordered phase. This zero-temperature transition is controlled by
tuning a high strength magnetic field. Target system 3 consists of Cooper
pairs in an array of connected superconductors. The transition from a
superconducting phase and an insulator phase is controlled by manipulating the
ratio of the Josephson coupling energy and the electrostatic interaction
energy. These analogues are summarised in Table 1.

| System | Boson | Phase transition control
---|---|---|---
Source | Cold atoms |  ^87Rb atom | Magnetic field/laser properties
Targets |  ^4He adsorbed in Vycor |  ^4He atom |  ^4He density
| Quantum dimer magnet | Dimer triplon | Magnetic field
| Josephson junction array | Cooper pair | Josephson energy and capacitance
Table 1: Comparison of analogue Bose-Hubbard systems.

###  2.3 Internal Validity of Analogue Quantum Simulations

The key methodological question now is how might scientists make valid
inferences about properties of the various target systems by probing the cold-
atom source system? How can the source-target inference be validated? The
first and most obvious step in such validation is for the scientists to ensure
that they are probing the source system in precisely the intended way. In
other words, the properties of the wavefunction implemented in the concrete
experiment should (approximately) match the properties of the intended
wavefunction; the simulation must be _internally valid_. Let us briefly
discuss techniques for internal validation here, before we move on to the core
of our argument regarding _external validation_ in the next section.222See
Franklin and Perovic (2016) and Winsberg (2019) for discussion of internal and
external validation in the context of experiments and classical computer
simulations respectively.

In the concrete case of the Bose-Hubbard simulator, internal validation is
most often achieved in the same way that conventional experiments are
internally validated. An elaborate ‘lab model’ of the concrete laboratory
system, including all known interactions and noise sources, is iteratively
constructed and validated through experimental probing. Since the experimental
system is so versatile, in that interactions can be switched on and off, and
since there are various methods available for probing the system (Bloch _et
al._ , 2008), individual ‘parts’ of the experiment can be individually
characterised. In the case of cold-atom systems this involves, for example,
characterising very precisely the properties of the BEC in the absence of an
optical lattice, validating the model parameters by probing the superfluid-
insulator transition, measuring the temperature of the system, characterising
the hopping terms, both nearest- and next-nearest neighbour, and
characterising the Feshbach resonance that is used to tune the interaction
strength (e.g. Höfer _et al._ , 2015).

A key challenge for such techniques is the complexity of quantum systems at
scale that prohibits the availability of classical verification. In short, the
complex quantum dynamics of such systems cannot be verified either
analytically or by classical computation due to their computational
intractability. This is typically addressed by comparing the model predictions
with experimental data in a classically tractable regime and inferring to
intractable regimes. For the Bose-Hubbard simulator, such comparisons show
excellent agreement even without free model parameters (Trotzky _et al._ ,
2012; Schreiber _et al._ , 2015). Provided that no additional sources of noise
occur when the individually benchmarked components of the experiment are
composed, the comparisons in the tractable regime provide a certificate for
the intractable regime. The resulting inference from classically tractable to
intractable regimes (e.g. Choi _et al._ , 2016) is an instance of the
application of a scale uniformity principle: behaviour at the scale of quantum
complexity is inferred from observed behaviour at the scale of analytic or
numerical computational complexity.333For more on the interplay between
inferences from independent uniformity principles, see (Evans and Thébault,
2020).

With the advent of versatile and universal quantum simulators and computers,
ways have been developed of both characterising directly and verifying the
features of the object of interest in an experimental scenario – the quantum
state, Hamiltonian, or process (Eisert _et al._ , 2020). But also for analogue
systems, methods for direct validation of the experimentally implemented
object of interest have been developed, including in particular the
identification of the Hamiltonian or Liouvillian parameters (Hangleiter _et
al._ , 2021; Samach _et al._ , 2021), benchmarking of Hamiltonian time-
evolution across the parameter range accessible in the experiment (Helsen _et
al._ , 2020; Derbyshire _et al._ , 2020; Shaffer _et al._ , 2021), and
fidelity estimation of a quantum state (Elben _et al._ , 2020). In another
vein, it has also been argued that analogue simulations might often be
insensitive to certain details of the experiment, for example due to slack in
the model space (Sarovar _et al._ , 2017), or because certain noise processes
affect both the simulator and the target in the same way (Cubitt _et al._ ,
2018).

Internally validated analogue simulation of the Bose-Hubbard model can be
applied as an _analogue quantum computation_ in order to compute abstract
properties of the Bose-Hubbard model itself (Hangleiter _et al._ , 2022). But
can we also probe the _physics_ of the various target systems in the analogue
simulation? After all, interest in the Bose-Hubbard model is motivated by the
fact that it describes a large variety of physical systems and their
superfluid-insulating transitions in the first place. In other words, can we
perform an _emulation_ of the target systems in the source system, similar to
an experiment?

##  3 Uniformity Principles in Analogue Quantum Simulation

###  3.1 Tokens, Types and External Validation

In order to ensure that the outcomes of an experiment on a particular physical
system are relevant to other physical systems with the same properties, we
need to _externally validate_ the experiment. Typically, conventional
experiments are performed with systems in mind that have the same, or a
similar, material constitution. Such systems are believed to behave similarly
when probed in the same circumstances. External validation then amounts to
ensuring that the specific lab system has the same material properties as the
target systems. More abstractly speaking, in an experiment a specific _token_
physical system is probed in order to learn about an entire _type_ of systems.
The inference from the token to the type is based on a _uniformity principle_
, which asserts that all systems of the same material constitution behave in
the same way when probed in the same circumstances.

In analogue experiments, by contrast, scientists aim for a system of one type
stand in for a system of another type, the latter of which importantly has a
distinct material constitution. In our case study, for instance, we have the
source system consisting of cold atoms and the target systems consisting of
Josephson junctions or Helium-4. It appears that by definition we cannot make
use of an intra-type uniformity principle between such systems since they are
materially distinct.

In order to establish that a system of one type can stand in for a system of
the other type, we would need to perform experiments in both systems in the
same setting and compare their outcomes. This would establish uniformity
between tokens of different types. However, the purpose of an analogue quantum
simulation is typically to probe the target system in a regime that is
experimentally inaccessible. How can we provide a reliable means for
justification of the relevant chain of inferences in such circumstances? One
way to achieve this would be to establish a specific _inter-type uniformity
principle_ between certain systems. But how could inter-type uniformity be
justified and which systems would fall under it? For intra-type uniformity
principles, the criterion was clear: it is the material constitution of the
systems. For inter-type uniformity principles (even assuming their existence)
this is less clear: Are we considering a uniformity principle between two
types? Should all tokens of the type, in all parameter regimes, be captured by
the uniformity principle?

###  3.2 Material Types, Empirical Types, and Universality Types

The natural implications of the discussion of the previous section is that in
the context of analogue quantum simulation we require uniformity principles
that cut across the boundaries of different types. The corresponding notion of
‘type’ is characterised by the material constitution of the systems. Let us
therefore define the notion of a _Material Type_ as follows.

Material Type

Two token systems are of the same _material type_ if they share the same
material composition as determined by the properties and spatial arrangement
of the constituent particles, atoms or molecules.

This is a simple and intuitive notion of type in that it fleshes out the
conceptual distinction that theoretical scientists and philosophers of science
would standardly draw upon. It is, however, a conceptualisation of type that
is not particularly well suited to the context of experimental science.
Consider the relevance of impurities within a sample for instance. Whether
such impurities in a source system are significant enough to render an
inference between source and target systems unreliable depends entirely on the
form of inference and the sensitivity of the experimental protocol. It might
be perfectly valid to treat two systems as of the same material type in the
context of one experimental inference despite a high level of impurities in
the target, say, but entirely invalid to treat the same two systems as of the
same material type in the context of another experimental inference.

The highly contextual nature of intra-type reasoning in experimental science
might thus prompt us to re-consider the focus on material constitution as the
basis for distinguishing types.444Here we take the motivations behind the
account of Bursten (2018) to be along broadly similar lines. What matters in
the context of an experimental inference is that the source and target
physical systems should behave similarly in similar situations. This motivates
us to define a notion of _Empirical Type_.

Empirical Type

Two token systems are of the same _empirical type_ , in a specified parameter
regime and with respect to a set of experimental prescriptions, if equivalent
implementations of the prescriptions in the parameter regime result in similar
measurement outcomes.

This notion of empirical type has a strong operationalist flavour but need not
be interpreted as requiring any commitment to operationalism in its
deployment. In particular, introducing the notion of empirical type in this
manner does not require us to _reduce_ the entire concept of type to
experimental operations. Rather, as we will see, it allows us to more
sensitively distinguish the differing notions of type that are deployed within
scientific reasoning. In particular, in a conventional experiment, the
uniformity reasoning deployed might reasonably be taken to implicitly rest on
the assumption that the notion of material type that is fundamental, and that
the notion of empirical type is a secondary implication. We can thus
understand the intra-type uniformity principles applied in conventional
experimentation built around the assumption that tokens of a material type are
also of the same empirical type.

Such reasoning assumes that all tokens of the same material type can be
described by a single theoretical model. Such a model can therefore be
validated by performing an experiment on a token system and applying the
intra-material-type uniformity principle. In order to justify the application
of the uniformity principle, the experiment needs to be externally validated.
In external validation, we ensure that the concrete token system we are
probing is in fact representative of the type we want to make an inference
about. In other words, for a material type, the similarity in material
constitution of the system needs to be established. A similarity in _nomic
behaviour_ is then assumed to lead to a similarity in empirical
behaviour.555According to the ‘better best system account of lawhood’ (Cohen
and Callender, 2009), it is similarity in empirical behaviour that leads to
the stipulation of a shared nomic behaviour. Given that the better best system
account balances natural kind ascriptions with economy and informativeness of
descriptive systems to generate laws optimised for predictive efficiency, we
see that there is a clear connection between external validation of material
constitution and ‘natural kind’ reasoning—one that warrants further
investigation. This is the essence of the uniformity reasoning that pertains
to the external validation of a conventional experiment.

Putting aside whether or not such an analysis of uniformity principles is
entirely adequate to the context of conventional experimental science, it is
clearly problematic in the context of analogue quantum simulation. That is, in
the context of such experimental practice, scientists clearly are not aiming
to justify an inference between source and target systems that are two tokens
of the same material type, and are thus not looking to establish similarities
in material constitution in order to establish nomic and empirical uniformity.
We might therefore seek to re-conceptualise the schema sketched above and
consider an _inter-material-type_ uniformity principle that would underlie the
reasoning at hand in place of the _intra-material-type_ uniformity principle.
To make this explicit consider the idea of a _Universality Type_ :

Universality Type

Two systems which are of different material types are of the same
_universality type_ if, in a specified parameter regime and with respect to a
set of experimental prescriptions, the behaviour displayed by the systems is
independent of their differences in material composition.

Notice that the difference between the notion of a universality type and an
empirical type is a subtle difference in emphasis. Whereas to belong to the
same empirical type two systems need to display the _same_ behaviour, to
belong to a universality type, that behaviour is merely required not to depend
on material composition. Other things being equal, membership in a
universality type thus implies membership in an empirical type, just as
membership in a material type implied – other things being equal – membership
in an empirical type.

However, a universality type provides us with a route to external validation:
an analogue quantum simulation might be validated on the basis of universality
arguments showing the _independence_ of the measurement outcomes on material
constitution between source and target. Such an argument would show that the
source and target systems belong to the same universality type, and thus,
other things being equal, being of the same empirical type on that
basis.666The sense of universality we are deploying here is a purposefully
broad one. That is, two systems may be of the same universality type in our
sense without being in the same ‘universality class’ in the Wilsonian sense of
being described by Hamiltonians that flow to the same infra-red fixed point
under renormalization group transformations. For analysis of the structure of
various types of universality arguments see Gryb _et al._ (2021) and
references therein. In essence, inferential work previously performed by
assumptions with regard to material types and laws of nature is now done by
the uniformity within the universality type. In each case the key step is to
establish target and source as members of the same empirical type, but in the
two cases this is achieved using a very different chain of reasoning.777The
form of argument that deploys universality types in support of source-target
inferences in the context of analogue experiments has been controversially
discussed in the recent literature on the epistemology of analogue
experimentation: see (Dardashti _et al._ , 2017, 2019; Thébault, 2019; Evans
and Thébault, 2020) for the case in favour of universality arguments as a
means for external validation in the context of analogue experimentation and
(Crowther _et al._ , 2019; Gryb _et al._ , 2021; Field, forthcoming) for more
sceptical commentary.

This suggests the question: can reasoning based on similarity as to empirical
type can be justified without appeal to material constitution _or_
universality arguments? Can we justify uniformity principles between empirical
types directly? Our goal in this work is to assess the possibility of
uniformity principles that operate directly at the level of empirical types. A
first lesson from universality types is that an independently established
confidence in the universality argument drives external validation. A lesson
from material types is that a physical assumption forms the basis of the
corresponding uniformity principle.

We argue that in the context of analogue quantum simulation there exists a
physical uniformity principle that is based on independently established
_empirical_ evidence. When successfully implemented, this strategy allows us
to validate the model of the target system in some empirically accessible
parameter regime and use such an independent uniformity principle to argue
that the model also applies in another parameter regime. In such circumstances
we can justifiably use the source system to probe the target system in an
empirically inaccessible regime.

###  3.3 Empirical Quantum Types

Let us then consider a physical uniformity principle that cuts across material
types based on independently established empirical evidence. This uniformity
principle gains its inferential power by leveraging the validity of quantum
theory in a well-characterised regime. The predictions of quantum theory have
been confirmed to extremely high precision and at scales ranging from the size
of the constituents of atoms to mechanical oscillators. Scientists can
reliably exploit quantum phenomena to build extremely precise clocks with an
accuracy of $10^{-14}$ Hz (Ludlow _et al._ , 2015), and measure ever so slight
signals due to gravitational waves using squeezed light (Aasi _et al._ ,
2013). In short, we are well justified to hold high confidence in the validity
of quantum theory, and knowledge of its applicability.

As we will argue, this confidence in quantum theory can be used to justify a
hybrid between our definitions of empirical and universality types, the notion
of an _Empirical Quantum Type_.

Empirical Quantum Type

Two token systems are of the same _empirical quantum type_ , in a given
parameter regime and with respect to given experimental prescriptions, if the
same quantum mechanical model can be deployed in that parameter regime to
provide an empirically adequate description of the systems for the
experimental prescriptions.

This definition allows for the possibility that the two systems at hand may be
of different material types since, as explicitly illustrated by our case
study, there clearly are cases in which the same quantum model can be deployed
to provide an empirically adequate description of systems with very different
material constitution. In general, what we would expect is that in the two
cases the quantum mechanical state of the systems can be adequately described
by the same wavefunction, that is, such that the properties of this
wavefunction when measured in the same setting result in approximately the
same outcomes for the two systems. While the material constitution of tokens
of a quantum type might differ, we are confident that all tokens are described
within the same _modelling framework_ across a certain parameter regime,
namely, quantum theory.

This constitutes a uniformity principle that can be exploited in inductive
inferences regarding individual tokens in a similar way as universality
arguments can. In contrast to universality arguments, which rely on _non-
empirical_ considerations, there is a concrete _empirical_ basis for this
uniformity principle. As we elaborate below, the quantum uniformity principle
allows for external validation of analogue quantum simulations employing
empirical quantum types in an inductive argument.

More specifically, in the context of analogue quantum simulation, the relevant
empirical quantum type is defined by an idealised _simulation model_
(Hangleiter _et al._ , 2022). Given that both the target and the source
physical systems are approximately described by the simulation model in a
certain parameter regime, their empirical properties in this parameter regime
will be approximately the same. Our most accurate description of the source
and target systems will be specific _system models_ that include all known
interactions and noise sources. Those are related to the simulation by de-
idealisation. We can think of all tokens of an empirical quantum type that
share the same material constitution, and therefore the same system model, as
a material sub-type of the empirical type.

This way of thinking about an empirical quantum type in the context of
analogue quantum simulation also provides a clear recipe for how to define
what we called ‘equivalent experimental prescriptions’ in Section 3.2. We can
specify an experimental prescription in terms of the idealised simulation
model that jointly and approximately describes all tokens of the empirical
quantum type. In other words, as long as there is a well-defined way in which
an experimental prescription can be specified and translated into equivalent
prescriptions for systems of different material constitution, this
prescription can figure in the definition of an empirical type. For tokens of
different material sub-type, we then simultaneously de-idealise the
experimental prescriptions in accordance with the de-idealisation to the
respective system model, giving rise to equivalent experimental prescriptions.

##  4 External validation of analogue quantum experiments

Now that we have established a theoretical framework for understanding the
inferential structure of external validation in conventional and analogue
experiments, let us return specifically to analogue quantum simulation. To be
explicit, let us assume that we want to perform an experiment on a source
quantum system $S$ in order to make an inference about another target quantum
system $T$ of a distinct material constitution. This analogue quantum
simulation is valid if both systems $S$ and $T$ are in the same quantum
empirical type, in that both systems are adequately described by the same
idealised quantum simulation model $M$, say, a Hamiltonian or a Liouvillian.
Let us assume that the experiment on system $S$ is internally valid, for
example using the methods described above in Section 2.3. We argue that this
analogue quantum simulation is also externally valid provided the following
conditions are satisfied.

  * (Q)

System $T$ is accurately described within the framework of quantum theory in a
certain parameter regime $P$.

  * (H)

System $T$ is accurately described by the idealised quantum simulation model
$M$ for some values of its parameters $P_{0}\in P$.

  * (T)

We have theoretical reasons to believe that $M$ accurately represents $T$ in
the parameter regime $P$.

We can then invoke the quantum uniformity principle (Q) together with the
usual uniformity principles to _inductively argue_ that:

  * (C)

System $T$ is accurately described by $M$ in the entire parameter regime $P$.

On a high level, the argument leverages a small piece of empirical knowledge
of the target system in a narrow regime as captured by condition (H) in order
to generalize the applicability of the model $M$ to a broad parameter regime
based on the broad quantum uniformity principle (Q). The condition (Q) can be
viewed as an epistemic aid to external validation of an analogue simulation,
because it significantly reduces the burden placed on external validation: If
one accepts the argument, it is sufficient to validate the simulation model
for _specific_ parameters and inductively extend the applicability of the
model. In contrast to material-type based reasoning, however, while the
conditions of the argument point to the validity of the conclusion, the
quantum uniformity principle does not directly posit the applicability of $M$.
This makes the generalizing inference broader and thus the argument – as one
would expect – weaker than standard experimental inference based on material-
type reasoning.

Given that the conditions (Q)+(H)+(T) are satisfied, we can then perform an
experiment on a quantum system of one material type to make a reliable
inference about a quantum system of another material type in a regime which is
inaccessible in the target system. Thus, the argument presented allows us to
make genuinely novel inferences about properties of concrete physical
phenomena in the target system that are not accessible through direct
experimentation.

How can we establish conditions (Q) and (H)? Let us begin with condition (H):
This condition can be established by a conventional experiment on the target
system or a token of the same material sub-type in the conventional sense of
an experiment. While the target system $T$ may be inaccessible in some
parameter regime of interest, it might be accessible in another regime that
can be experimentally probed. Moreover, given that we are in the realm of
applicability of quantum theory, in this regime we also want to be able to
compare the predictions of quantum theory with experimental outcomes, so it is
advantageous to perform an experiment in the computationally tractable regime.
To establish condition (Q) we can invoke our confidence in the validity of
quantum theory as a whole. What remains is to validate that the target system
is also in the parameter regime in which we are confident of the quantum
uniformity principle. This is possible, for example, by means of experiments
on the target system in regimes that are experimentally accessible.
Alternatively, we may be able to make a non-empirical assessment of the range
of validity, drawing on independently validated information.

###  4.1 External validation of Bose-Hubbard analogue simulations

Our framework for understanding the external validity of analogue quantum
experiments by design can be applied to the context of the Bose-Hubbard
analogue simulations outlined in Section 2.2. The optical lattice system is
our source system S, and the superfluid helium, quantum dimer magnet, and
Josephson junction array are our target systems $T_{1}$, $T_{2}$, $T_{3}$,
respectively. We take each of these systems to be described by the same
idealised quantum simulation model $M$ – the Bose-Hubbard model – in the right
parameter regime $P$. And we take there to be high confidence that the
relevant probing experiments on the optical lattice system are internally
valid.

According to our framework, whether the optical lattice analogue quantum
simulation of the three target systems counts as externally valid turns on
whether each system $T_{i}$ is accurately described (Q) within the framework
of quantum theory within parameter regime $P$, and (H) by the Bose-Hubbard
model for some subset of parameters within $P$. Condition (H) is established
by conventional experimentation on the target systems. For the Bose-Hubbard
systems, experiments were performed wherein the superfluid-Mott insulator
transition was observed; recall §2.1. This is a typical condition that would
need to be met by any analogue experiment.

Condition (Q), however, is in a sense the key to the external validity of
analogue _quantum_ experiments. There is a multitude of independent lines of
evidence that each of the target systems are well described by quantum theory
in the appropriate parameter regime. As we mentioned above, this reduces the
inferential burden on external validation and allows for genuinely novel
inferences about properties of inaccessible concrete phenomena in the target
system. It is thus by employing reasoning along these lines that we gain
confidence that the optical lattice analogue simulation tells us something
about the nature of the superfluid-insulator phase transition in the variety
of target systems.

Once we have independent confidence that both conditions (Q) and (H) hold,
their combination allows us to argue inductively that the target systems
$T_{i}$ are described by the Bose-Hubbard model in the entire applicable
parameter regime. This inference underpins the claims typical of analogue
quantum simulations that probing the accessible behaviour of the source system
can be taken to probe the inaccessible behaviour of the target system.

It is important to note the limited parameter regime in which each of the
target systems will be accurately described by quantum theory. At a certain
level of coarse-grained abstraction, the ^4He fluid, the magnetic material,
and the Josephson junction array will all behave classically. We do not expect
the analogue quantum simulation to provide evidence for behaviour in this
extended parameter regime. But at the appropriate fine-grained description –
at which one can generate confidence in the quantumness condition (Q) – we can
then infer the relevant Bose-Hubbard model to be an accurate description.
Insofar as these two conditions are verified independently, this provides
external validation of these analogue quantum simulation.

Even more strongly, because we can validate the applicability of Bose-Hubbard
dynamics in the target systems in some tractable regime, we are justified in
making inferences about the Bose-Hubbard behaviour of those systems in
intractable regimes based on the behaviour observed in the analogue simulation
experiments. Such arguments in effect justify simultaneous de-idealisation of
a single abstract quantum model to source and target system within a
designated range of applicability and in so doing provide justification for
reasoning based upon regularity within empirical types.888There is a natural
connection here to model based reasoning as discussed in general by Bokulich
(2017) and in the context of analogue quantum simulation by Hangleiter _et
al._ (2022)

We thus reach the remarkable conclusion that despite the fact that the four
analogue systems are instances of wholly different material constitutions, we
expect them to obey structurally similar phase space and critical point
dynamics on account of the strength of the analogue simulation. In a sense,
what we see is that an inter-type uniformity principle can become an intra-
empirical-quantum-type uniformity principle, which then underpins the external
validity of the analogue experiments. This then replicates the inferential
structure of conventional experimentation in the physical sciences where
intra-type uniformity underpins the external validation.

###  4.2 Universality-based external validation reconsidered

Before we conclude, it is worthwhile to re-examine the connection between
external validation of analogue simulation via empirical quantum types and via
universality arguments. Above we noted that analogue quantum simulation is
applied in a variety of contexts where appeal to universality arguments is not
part of the inferential process. As such, there is a clear demand for an
analysis of the forms of empirical uniformity principles applied by scientists
to infer from source system behaviour to target system behaviour in this more
general context. This is what our analysis of empirical quantum types is
designed to achieve.

One might ask, however, whether an analogous argument is possible for
universality types. In particular, there are examples of empirically grounded
universality arguments based on renormalization-group techniques which can be
applied in the context of analogue quantum simulations (see e.g. Prüfer _et
al._ (2018); Anthore _et al._ (2018). Broadly speaking, universality based
upon renormalization-group techniques has – like quantum mechanics – been
abundantly observed in nature and plausibly we can take its empirical status,
within the its relevant domain, to be comparable to that of quantum theory.
Within our framework, we are then in a situation in which we both have certain
models of the source and target system and an empirically well-established
inter-material-type uniformity principle.

Such an empirically well-established physical universality property then
allows us to directly validate membership in an empirical type, in the same
vein as the argument above. Analogously to the conditions (Q) and (H), to
achieve this requires us to simultaneously satisfy two conditions. First, we
need to acquire empirical evidence in the validity of the respective models.
In particular, we require 1 detailed knowledge of the mirco-structure of the
source and target. Second, we require empirical evidence for the applicability
of relevant universality arguments in terms of 2 empirical access to the macro
behaviour of the source and target.

This conclusion establishes a compelling connection with the work of Field
(forthcoming) who argues very plausibly that precisely the conditions 1 and 2
should be applied as individually sufficient conditions for an analogue
simulation to be ‘universality-argument-apt’. That is, for the situation to be
such that a universality argument, together with experiment on a source
system, is able to achieve significant confirmatory power with regards to
hypotheses about the target system. Our analysis thus returns a similar result
to Field for external validation of (empirical) universality types, except we
take the conditions to be jointly sufficient.

Clearly, as Field herself notes, such circumstances are unlikely to obtain in
the context of exotic examples of analogue simulation such as those found in
analogue gravity. However, in the context of accessible matter systems where
reliable universality arguments based upon renormalization-group techniques
are available, joint satisfaction of the conditions is highly plausible.

##  5 Discussion

This paper has provided the first philosophical investigation of the
epistemology of the novel experimental scientific practice of ‘wavefunction
engineering’. Wavefunction engineering relies upon regularity between quantum
systems which exemplify the same empirical type, despite having different
material constitution. We have argued that, in such contexts, limited
empirical access to both source and target system can be leveraged into
external validation of analogue simulations by the independently established
confidence in quantum theory.

One might wonder where the bulk of the work is done in this argument: On the
one hand there is the underlying, but broad, uniformity principle, and on the
other hand there is the specific, but narrow, empirical evidence for the
validity of the model due to direct observation. Specifically, one might ask
whether the uniformity principle is adding a quantitative or a qualitative
difference to the argument. After all, it is standard practice to confirm
models by performing experiments in restricted parameter regimes. We argue
that the difference is qualitative – we would not be able to conclude the
broad validity of the simulation model in _both_ source and target system
across the entire parameter regime of interest if we were not very confident
in the validity of the modelling framework. In contrast, in a standard
experiment, we are confident that the same model is valid if two systems have
the same material constitution.

The epistemology of analogue quantum simulation in cases like those we have
considered can be compared and contrasted with that of analogue
experimentation in the context of remote or entirely inaccessible phenomena
such as analogue gravity (Dardashti _et al._ , 2017), wherein justificatory
arguments are framed in terms of the universality of phenomena across
different material types. It remains to be seen, however, how strongly this
distinction should be taken. On the one hand, as argued by Winsberg (2009,
2010) in the context of experimentation and classical computer simulation,
plausibly if we want to characterize the difference between two methods we
should not focus on what objective relationship actually exists between the
object of an investigation and its target. Rather, what distinguishes
different methods is the character of the argument given for the legitimacy of
the inference from object to target and the character of the background
knowledge that grounds that argument. On such a view the distinction between
wavefunction engineering and analogue experimentation based on universality
would be a robust one as the type of argument to support the inference is
distinct. However, on the other hand, at an ontological level, the distinction
between _intra_ -empirical-type uniformity and inter-material-type uniformity
does not seem to be grounded in a clean or straightforward distinction. What
this means for the epistemology of analogue gravity experiments remains to be
seen.

Similarly, we can compare analogue simulation to both standard experimentation
and simulation. Taking again Winsberg’s view as the basis, the distinction
between simulation and experimentation is grounded in what kind of evidence we
refer to when justifying inferences: Plausibly, one might think that our
arguments for the validity of a computer simulation are model-based, whereas
our arguments for the validity of an experiment are material-type based. On
this view, given our empirical-quantum-type argument, analogue quantum
simulation is a practice that is genuinely intermediate between simulation and
experimentation. Its justification is grounded both in model-based reasoning
wherein source and target system are described by the same quantum-mechanical
model, and the nomic argument that both systems are physically similar in the
sense that they are both quantum-mechanical. Possibly the issue is that the
epistemology and ontology of simulation and experimentation cannot be
separated. Rather, since it is the mode of de-idealisation that is different
in the two cases, we should not be trying to differentiate between what there
is and what we know. As the practice of wavefunction engineering and analogue
experimentation continue to thrive, such issues will become of increasing
importance, and thus warrant further investigation.

## Acknowledgements

We wish to thank Ivan Kassel and Andrew White for illuminating discussions on
this topic. We are also grateful to the audience of the Symposium ‘From Bosons
to Markets to Black Holes: New Prospects for Analogical Reasoning’ at the PSA
2021 meeting in Baltimore. P.W.E. acknowledges support from the University of
Queensland and the Australian Government through the Australian Research
Council (DE170100808) and from the Foundational Questions Institute and Fetzer
Franklin Fund (FQXi-RFP-CPW-2019), a donor advised fund of Silicon Valley
Community Foundation. D.H. acknowledges financial support from the U.S.
Department of Defense through a QuICS Hartree fellowship.

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