Efficient tomography of a quantum many-body system
B. P. Lanyon
†
,
1, 2, ∗
C. Maier,
1, 2, †
M. Holzäpfel,
3
T. Baumgratz,
3, 4, 5
C. Hempel,
2, 6
P. Jurcevic,
1, 2
I. Dhand,
3
A. S. Buyskikh,
7
A. J.
Daley,
7
M. Cramer,
3, 8
M. B. Plenio,
3
R. Blatt,
1, 2
and C. F. Roos
1, 2
1
Institut für Quantenoptik und Quanteninformation,
Österreichische Akademie der Wissenschaften,
Technikerstr. 21A, 6020 Innsbruck, Austria
2
Institut für Experimentalphysik, Universität Innsbruck,
Technikerstr. 25, 6020 Innsbruck, Austria
3
Institut für Theoretische Physik and IQST,
Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany
4
Clarendon Laboratory, Department of Physics,
University of Oxford, Oxford OX1 3PU, United Kingdom
5
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
6
ARC Centre for Engineered Quantum Systems,
School of Physics, The University of Sydney,
Sydney, New South Wales 2006, Australia.
7
Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, UK
8
Institut für Theoretische Physik, Leibniz Universität Hannover, Hannover, Germany
(Dated: July 7, 2017)
∗
Electronic address: ben.lanyon@uibk.ac.at
†
These authors contributed equally to this work.
1
Quantum state tomography (QST) is the gold standard technique for obtaining an estimate
for the state of small quantum systems in the laboratory [1]. Its application to systems
with more than a few constituents (e.g. particles) soon becomes impractical as the effort
required grows exponentially with the number of constituents. Developing more efficient
techniques is particularly pressing as precisely-controllable quantum systems that are
well beyond the reach of QST are emerging in laboratories. Motivated by this, there is
a considerable ongoing effort to develop new state characterisation tools for quantum
many-body systems [2–11]. Here we demonstrate Matrix Product State (MPS) tomography
[2], which is theoretically proven to allow the states of a broad class of quantum systems to
be accurately estimated with an effort that increases efficiently with constituent number. We
use the technique to reconstruct the dynamical state of a trapped-ion quantum simulator
comprising up to 14 entangled and individually-controlled spins (qubits): a size far beyond
the practical limits of QST. Our results reveal the dynamical growth of entanglement and
description complexity as correlations spread out during a quench: a necessary condition for
future beyond-classical performance. MPS tomography should therefore find widespread
use to study large quantum many-body systems and to benchmark and verify quantum
simulators and computers.
An MPS is a way of expressing a many-particle wave function which, for a broad class of physical
states, offers a compact and accurate description with a number of parameters that increases only
polynomially (i.e. efficiently) in system components. [12]. MPS tomography recognises that
the information required to identify the compact MPS is typically accessible locally; that is, by
making measurements only on subsets of particles that lie in the same neighbourhood [2]. In
such cases, the total effort required to obtain a reliable estimate for the state in the laboratory
increases at most polynomially in system components [2, 6]. States suited to MPS tomography
and its generalisations to higher dimensions [2] include those with a maximum distance over which
significant quantum correlations exist between constituents (locally-correlated states) e.g. the 2D
cluster states—universal resource states for quantum computing—and the ground states of a broad
class of 1D systems [13–15]. We find that MPS tomography is also well-suited to characterise
the states generated during the dynamical evolution of systems with short-ranged interactions, as
found in many physical systems (Methods and [16]).
Consider an N-component quantum system initially in a product state (or other locally-
2
correlated state) in which interactions are abruptly turned on. In the presence of finite-range
interactions, information and correlations spread out in the system with a strict maximum group
velocity [17–19]. Therefore, after a finite evolution time, there is a maximum distance over which
correlations extend in the system (the correlation length, L), beyond which correlations decay
exponentially in distance. The information required to describe the state is largely contained in
the local reductions: the reduced states (density matrices) of all groups of neighbouring particles
contained within L. In 1D systems, such states can be described by a compact MPS [15, 20]
and, to identify and certify the total N-component MPS, the experimentalist need only perform
the measurements required to reconstruct the local reductions [16]. Each local reduction can be
determined by full QST, requiring measurements in at most 3
L
bases. Since the number of local
reductions increases only linearly in N, the total number of measurement bases scales efficiently
in this parameter. The local reduction estimates are passed to a classical algorithm which finds
an MPS estimate in a time polynomial in N [2, 7] (see Figure 1). We find that the total num-
ber of measurements required to obtain a desired fidelity for the state reconstruction also scales
efficiently in N (Methods).
Our strategy is not restricted to 1D systems nor to those with strictly finite range interactions. In
any of those cases, we benefit from the fact that MPS tomography and its natural generalisations
to higher spatial dimensions [2] make no prior assumptions about the form of the state in the
laboratory (e.g. that it is pure or well-described by a compact MPS), because the state estimate
can be certified: an efficient assumption-free lower bound on the fidelity with the laboratory state
ρ
lab
is provided [2]. For example, the correlation length L need not be known a priori. If, after
measurements on k-sites, the certified minimum fidelity F
k
c
between the MPS estimate |ψ
k
c
i and
the state in the laboratory ρ
lab
is deemed not high enough, then one can try again for larger k.
Generalisation of our method to higher spatial dimensions and to mixed-state estimates using
matrix product operators [6, 7] is possible, although no general certification method is currently
known for mixed states [29].
For finite range interactions, the correlation length L can increase at most linearly in time as
entanglement grows and spreads out in the system, demanding exponentially growing number of
measurements to estimate each local reduction [21, 22]. This puts practical limits on the evolution
time until which the system state can be efficiently characterised: once correlations have spread out
over the whole system the effort for MPS tomography becomes the same as full QST. MPS tomog-
raphy is able to verify evolution towards classically-intractable regimes: as the system evolves, the
3
size of the local reductions required to obtain an accurate pure MPS description should continue
to increase (as seen in our data).
Our quantum simulator consists of a string of trapped
40
Ca
+
ions. In each ion j=1 . . . N, two
electronic states encode a spin-1/2 particle. Under the influence of laser-induced forces, the spin
interactions are well described by an ‘XY’ model in a dominant transverse field B, with Hamil-
tonian H
XY
=~
P
i< j
J
i j
(σ
+
i
σ
−
j
+σ
−
i
σ
+
j
) + ~B
P
j
σ
z
j
. Here J
i j
is an N × N spin-spin coupling matrix,
σ
+
i
(σ
−
i
) is the spin raising (lowering) operator for spin i and σ
z
j
is the Pauli Z matrix for spin j.
Interactions reduce approximately with a power-law J
i j
∝ 1/|i − j|
α
with distance |i − j|. Here
1.1<α<1.6, for which the predominant feature of spreading wave packets of correlations is never-
theless evident [23–25]. Applying MPS tomography to study complex out-of-equilibrium states,
generated by interactions that are not strictly finite-range, represents a most stringent test of its
scope of application.
The largest application of full QST was for a simple 8 qubit W-state, employing measurements
in 6561 different bases taken over ten hours [26]. We begin experiments with 8 spin (qubit) quench
dynamics, and accurately reconstruct complex 8-spin entangled states using measurements in 27
bases taken over ten minutes. We measure in sufficient bases to reconstruct all k-local reductions of
individual spins (k = 1), neighbouring spin pairs (k = 2) and spin triplets (k = 3), during simulator
evolution starting from the initial highly-excited Néel state |φ(0)i = |↑, ↓, ↑, ↓ ...i (Methods). The
local measurements directly reveal important properties: single-site ‘magnetisation’ shows how
spin excitations disperse and then partially refocus (Figure 2a); in the first few milliseconds, strong
entanglement develops in all neighbouring spin pairs and triplets, then later reducing, first in pairs
then in triplets, consistent with correlations spreading out over more spins in the system (Figure
2c–d).
Certified fidelity lower bounds F
k
c
≤ hψ
k
c
|ρ
lab
|ψ
k
c
i from MPS tomography during the 8-spin
quench are shown in Fig 3a. The results closely match an idealised model where MPS tomography
is applied to exact local reductions of the ideal time-evolved states |φ(t)i(Methods). Measurements
on k = 1 sites at t = 0 yield a certified MPS state reconstruction |ψ
1
c
i, with F
1
c
= 0.98 ± 0.01 and
|hψ
1
c
|φ(0)i|
2
= 0.98, proving that the system is initially well described by a pure product Néel state.
As expected, the fidelity lower bounds based on single-site measurements rapidly degrade as the
simulator evolves, falling to 0 by t = 2 ms. Nevertheless, an accurate MPS (pure-state) description
is still achieved by measuring on larger (k = 2) and larger (k = 3) reduced sites. The model
fidelity bounds F
3
c
begin to drop after t = 2 ms, consistent with the time at which the information
4
wavefronts are expected to reach next-nearest-neighbours (light-like cones, Figure 2a), allowing
for correlations beyond 3 sites to develop. Measurements on k = 3 sites reveal an MPS description
with more than 0.8 fidelity up to t = 3 ms, before the lower bound rapidly drops to 0 at 6 ms.
This is consistent with the model and the entanglement properties measured directly in the local
reductions (Figure 2b–c): At t = 3 ms, entanglement in spin triplets maximises before reducing
to almost zero at 6 ms, as correlations have then spread out to include more distant spins. Beyond
t = 3 ms it becomes increasingly difficult to uniquely distinguish (and certify) the global state
based on 3-site local reductions (although the estimate can still be a good description).
The data in Figure 3a clearly reveal the generation and spreading-out of entanglement during
simulator evolution, up to 3–4 ms, and are consistent with this behaviour continuing beyond this
time. To confirm this, it would be necessary to measure on increasingly large numbers of sites,
demanding measurements that grow exponentially in k. That the amount of entanglement in the
simulator is growing in time can be seen from the inset in figure 3a: the half-chain entropies of
the certified MPSs |ψ
3
c
i are seen to grow as expected for a sudden quench [27]. For all times at
which F
3
c
> 0 (except t = 0), the pure MPS-reconstructed states |ψ
3
c
i are non-separable across all
partitions.
Figure 3b–c compares spin-spin correlations (‘correlation matrices’) present in |ψ
3
c
i at t = 3 ms
(F
3
c
> 0.84 ±0.05), with those obtained directly in the lab via additional measurements. The certi-
fied MPS captures the strong pairwise correlations in the simulator state and correctly predicts the
sign and spatial profile of correlations beyond next-nearest neighbour: that is, of state properties
beyond those measured to construct it (beyond k = 3).
Figure 4 presents results from a 14 spin quench: far beyond the practical limits of full QST. Full
QST on 14 spins would require measuring in more than 4 million bases. We reconstruct a certified
MPS estimate using only 27 local measurement bases. At t
14
= 4 ms, strong entanglement, in
neighbouring pairs and triplets, has developed right across the system. Measurements on 3 sites,
at t
14
yield an MPS estimate |ψ
3
c
i with a certified minimum fidelity of F
3
c
= 0.39 ± 0.08. Since F
k
c
are only lower bounds, it is natural to ask exactly what the state fidelity is. Using the estimated 14-
spin MPS state |ψ
3
c
i from MPS tomography, we perform Direct Fidelity Estimation (DFE) [4, 5]
with the experimentally generated state using an additional set of measurements, obtaining a result
of 0.74 ± 0.05 (Methods and FIG. S10).
Clearly MPS tomography provided an accurate estimate of the 14-spin simulator state, and
the fidelity lower bound of F
3
c
= 0.39 ± 0.08 is correct. However, the bound is conservative and
5
![FIG. 1: Generation and efficient characterisation of locally-correlated quantum states. a. Quantum spins fixed on a 1D lattice and initialised into some separable pure state. Finite-range spin-spin interactions are then abruptly turned on. In the subsequent dynamics, quantum correlations spread out with a maximum group velocity [17–19, 23, 28], producing light-like cones (grey arrows, only a few are shown) and a locallycorrelated entangled state. b. After the particular evolution time shown, quantum correlations have spread to neighbouring spin triplets (not all shown). The established correlation length is L = 3. The total N-spin state can be accurately described by a compact MPS, efficient in N. The correlation length increases at most linearly in time. c. To obtain an accurate MPS estimate for the state in the laboratory, the experimentalist need only perform sufficient measurements to reconstruct all N − L + 1 neighbouring spin triplet reduced density matrices (local reductions). At any given evolution time, the experimental effort therefore increases linearly in spin number N.](/figures/fig-1-generation-and-efficient-characterisation-of-locally-24xg8prv.webp)
