Quantum Process Tomography of Unitary Maps from Time-Delayed Measurements

Irene Lopez Gutierrez∗ and Felix Dietrich†

Technical University of Munich, Department of Informatics, Boltzmannstraße 3, 85748 Garching, Germany

Christian B. Mendl‡

Technical University of Munich, Department of Informatics,Boltzmannstraße 3, 85748 Garching, Germany and

Technical University of Munich, Institute for Advanced Study, Lichtenbergstraße 2a, 85748 Garching, Germany

Quantum process tomography conventionally uses a multitude of initial quantum states and thenperforms state tomography on the process output. Here we propose and study an alternativeapproach which requires only a single (or few) known initial states together with time-delayed mea-surements for reconstructing the unitary map and corresponding Hamiltonian of the time dynamics.The overarching mathematical framework and feasibility guarantee of our method is provided by theTakens embedding theorem. We explain in detail how the reconstruction of a single qubit Hamil-tonian works in this setting, and provide numerical methods and experiments for general few-qubitand lattice systems with local interactions. In particular, the method allows to find the Hamiltonianof a two qubit system by observing only one of the qubits.

I. INTRODUCTION

System identification refers to the estimation of the dy-namics of a system from measurements of its character-istics. Its quantum analogue, quantum process tomogra-phy (QPT), is essential for the realization and testing ofquantum devices [1–3], as a benchmarking tool for quan-tum algorithms [4, 5], and in general for understandingthe inner workings of a quantum system [6, 7]. However,textbook algorithms for QPT [8] might be difficult torealize in laboratory settings due to the required prepa-ration of many initial states and observation of the com-plete target system.

In this work we focus on the unitary time evolution ofa closed quantum system. We propose and investigate anapproach based on measurements with different time de-lays, which should be easily realizable in lab experiments.Our main contribution are algorithms and numerical pro-cedures for identifying the corresponding time evolutionoperator, and thus indirectly the quantum Hamiltonian.Intriguingly, we can identify the operator for the entiresystem even if the measurements are restricted to a sub-system (using two known initial quantum states). TheTakens embedding theorem, discussed in Sect. III, pro-vides an overarching mathematical foundation and feasi-bility guarantee for our approach. Concretely, the math-ematical framework starts with a manifoldM, which wetake to be the special unitary group of time evolutionoperators. Given an initial quantum state, a time stepmatrix U ∈ M then determines the measurement aver-ages at a sequence of time points, see Fig. 1. The Takenstheorem states that the map from M to this measure-ment vector is actually a smooth embedding (under cer-

∗Electronic address: irene.lopez@tum.de†Electronic address: felix.dietrich@tum.de‡Electronic address: christian.mendl@tum.de

U = e−iH∆t

M

|ψ〉 U Uγ U (γ2) · · ·U (γq)

t/∆t

〈ψ(t)|M |ψ(t)〉

1 γ γ2 γq· · ·

FIG. 1: Schematic illustration of the map from a unitarytime step operator U to a vector of measurement averages atdifferent time delays.

tain conditions), which then allows us to reconstruct U .

Related to the present work, a recent approach forquantum process tomography using tensor networks asa parametrization of the quantum channel was able toachieve high accuracies for systems of up to 10 qubits [9].Furthermore, in [10] the authors derive a lower bound inthe number of POVMs required to fully characterize aunitary or near-unitary map. Regarding the related taskof quantum state tomography, several methods, somebased on machine learning techniques, have been devel-

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oped [11–14]. However, except for [14], these works donot explicitly take advantage of the time trajectory of thequantum system as envisioned here. Gate Set Tomogra-phy [15] does not need pre-calibrated quantum states,but relies on very long time series.

Our work is closely related to earlier research on theidentification of quantum Hamiltonians from time se-ries [16, 17]. The authors obtain all degrees of freedom ofthe Hamiltonian with known structure by solving a sys-tem of equations, mostly in the presence of noise. Morerecently, neural networks have been used for identifica-tion, even from experimental data [18–20]. The minimalnumber of observables needed to identify the Hamilto-nian is not addressed, and we provide it here with thelink to Takens theorem. In our work, the choice of initialstate is not as important, as we will demonstrate.

Takens theorem has been fundamental to several sys-tem identification methods for general dynamical systemsalready [21], recently also involving machine learning [22–25], with a focus on PDE [26, 27] or special structure suchas (classical) Hamiltonian dynamics [28, 29]. The iden-tification of a unitary map from measurement data hasbeen discussed by Koopman and von Neumann [30, 31],work that has been revived in a data-driven context inthe last twenty years and extended to dissipative sys-tems [32–34]. The connection to quantum systems andtheir special structure and challenges has not yet beenaddressed in the work cited above, however.

II. PHYSICAL MODEL

We assume throughout that the quantum HamiltonianH is time-independent and will investigate two settings:(i) a few-qubit system and H a general dense Hermitianmatrix, and (ii) a (tight binding) Ising-type model on atwo-dimensional lattice Λ, as widely studied in condensedmatter physics. For concreteness, the physical system forcase (ii) consists a local spin degree of freedom at eachlattice site. The Hamiltonian is then defined as

H = −∑〈j,`〉

Jj,` σzjσ

z` −

∑j∈Λ

~hj · ~σj , (1)

where Jj,` ∈ R and ~hj ∈ R3 are local parameters defin-ing the interaction strength and the external field, re-spectively. σαj for α ∈ {x, y, z} is the α-th Pauli matrix

acting on site j ∈ Λ, and ~σj = (σxj , σyj , σ

zj ) the corre-

sponding Pauli vector. The first sum in (1) runs overnearest neighbors on the lattice. We take Λ to containa finite number n of sites, and assume periodic bound-ary conditions. The site-dependent parameters allow tosimulate disorder. Fig. 2 illustrates the interaction andexternal field terms of H on a two-dimensional lattice.The precise form of H is not important for the recon-struction as long as the time dynamics it generates canbe well approximated by a quantum circuit, for examplevia Trotterization. We assume that the overall structure

of H is known, and our task is to determine the numericalvalues of the parameters.

FIG. 2: Visualization of the quantum Hamiltonian in Eq. (1)on a two-dimensional lattice, with J the interaction strength

and ~h the external field.

For later reference, we state the unitary time evolutionoperator at time t ∈ R based on the Schrodinger equation(in units of ~ = 1):

U(t) = e−iHt . (2)

The solution to the Schrodinger equation for an initialwavefunction (statevector) ψ ∈ CN is then ψ(t) = U(t)ψ.

III. TAKENS EMBEDDING FRAMEWORK

The theorems of Takens and Ruelle [35, 36], based onthe embedding theorems of Whitney [37], are the theo-retical foundations of time-delay embedding we use here.

The general idea we employ in this paper is to “em-bed” the manifold of unitary matrices (describing thedynamics of a quantum system) into the space of mea-surement trajectories. We first state the mathematicaltheorem, and then discuss the specialization for quan-tum time evolution. Let k ≥ d ∈ N, and M ⊂ Rk bea d-dimensional, compact, smooth, connected, orientedmanifold with Riemannian metric g induced by the em-bedding in k-dimensional Euclidean space. Note thatthis setting is sufficient for our presentation, but is morerestrictive than allowed by the results cited below.

Together with the results from Packard et al. [38] andAeyels [39], the definitions and theorems of Takens [36]describe the concept of observability of state spaces ofnonlinear dynamical systems. A dynamical system is de-fined through its state space (here, the manifoldM) anda diffeomorphism φ :M→M.

Theorem 1. Generic delay embeddings For pairs(φ, y), φ : M → M a smooth diffeomorphism andy : M → R a smooth function, it is a generic property

3

that the map Φ(φ,y) :M→ R2d+1, defined by

Φ(φ,y)(x) =(y(x), y(φ(x)), . . . , y(φ ◦ · · · ◦ φ︸ ︷︷ ︸

2d times

(x)))

(3)

is an embedding ofM; here, “smooth” means at least C2.

Genericity in this context is defined as “an open anddense set of pairs (φ, y)” in the C2 function space. Openand dense sets can have measure zero, so Sauer et al. [40]later refined this result significantly by introducing theconcept of prevalence (a “probability one” analog in infi-nite dimensional spaces). See [41] for similar results withstochastic systems.

Let N denote the quantum Hilbert space dimension.In our context,M is the special unitary group SU(N) ⊂CN×N when identifying C ' R2. In particular, M (withunderlying field R) has dimension d = N2−1. In physicalterms, the elements of SU(N) are the unitary time evo-lution matrices in Eq. (2). We assume that the Hamilto-nian H is traceless, since adding multiples of the identityto H leads to a global phase factor in the time evolu-tion, which is unobservable in subsequent measurementsas envisioned here. We fix a time step ∆t, which wemay (without loss of generality) absorb into H, and setU = e−iH in the following.

Now define the diffeomorphism φ via a scaling of H bya factor γ > 0, γ 6= 1, such that

φ(U) = e−iγH = Uγ . (4)

Regarding y, fix a randomly chosen initial quantumstate ψ ∈ CN and an observable (Hermitian matrix) M .Now let y compute the corresponding expectation value:

y :M→ R, y(U) = 〈ψ|U†MU |ψ〉 . (5)

With these definitions, the output of the map Φ(φ,y) inEq. (3) becomes

Φ(φ,y)(U) =(〈ψ| eiHM e−iH |ψ〉 ,

〈ψ| eiγHM e−iγH |ψ〉 ,

. . . , 〈ψ| eiγ2dHM e−iγ

2dH |ψ〉), (6)

physically corresponding to measurements at time pointstq = γq for q = 0, . . . , 2d.

The main point here is the prospect to identify thetime step matrix U , and thus indirectly the Hamiltonian,based on a single measurement time trajectory, under theassumption that the initial quantum state is known. Ascaveat, the relation U = e−iH∆t determines the eigenval-ues of H only up to multiples of 2π/∆t. Moreover, themap Φ(φ,y) might not be one-to-one, in the sense that twodifferent unitary matrices give rise to the same measure-ment trajectory. We discuss such a case in more detailin the following. Such issues can be avoided in practiceby additional assumptions on the structure of H.

IV. NUMERICAL METHODS

Throughout this work, we assume that it is feasibleto reliably prepare a single (or when indicated several)known initial state(s) ψ ∈ CN . To demonstrate the gen-eral applicability of our methods, ψ is chosen at randomin the following algorithms and numerical simulations.Specifically, the entries of ψ before normalization areindependent and identically distributed (i.i.d.) randomnumbers sampled from the standard complex normal dis-tribution.

A. Reconstruction algorithm for a single-qubitsystem

x

y

z

~h

−~h′

~r

(a) trajectory on Bloch sphere

κ=λ+α22+λ-α32

norm constraint

-1.5 -1. -0.5 0.5 1. 1.5α2

-1.5

-1.

-0.5

0.5

1.

1.5

α3

(b) conditions on α2 and α3 coefficients

FIG. 3: (a) The Bloch sphere picture of the time dynamicseffected by a Hamiltonian is a classical rotation. For a singleobservable, the reconstruction of H is not unique: e.g., the

two trajectories for H = ~h · ~σ and H ′ = ~h′ · ~σ result in thesame 〈Z〉 expectation values. (b) The system of equations tobe solved for the reconstruction admits four solutions.

We now describe how to reconstruct a single-qubitHamiltonian from a time series of measurements. TheBloch sphere picture [8] provides a geometric perspective

4

and insight into single-qubit quantum states and opera-tions. The Bloch vector ~r ∈ R3 associated with ψ ∈ C2

is defined via the relation

|ψ〉 〈ψ| = 1

2(I + ~r · ~σ). (7)

For pure states as considered here, ~r is a unit vector.We can parametrize any single-qubit Hamiltonian (up tomultiples of the identity map, which are irrelevant here)as

H = ~h · ~σ (8)

with ~h ∈ R3. The corresponding time evolution operatoris the rotation

U(t) = e−iHt = cos(ωt/2)I − i sin(ωt/2)(~v · ~σ) ∈ SU(2),(9)

when representing ~h = ω~v/2 by a unit vector ~v ∈ R3 andω ∈ R. On the Bloch sphere, this operator correspondsto a classical rotation about ~v, as illustrated in Fig. 3aand described by Rodrigues’ rotation formula (θ = ωt):

Uθ,~v ~r = cos(θ)~r+sin(θ)(~v×~r)+(1−cos(θ))(~v ·~r)~v. (10)

Uθ,~v ∈ SO(3) is a rotation matrix (parametrized by θand ~v) applied to ~r. In the above context of Takensembedding with U = U(∆t), we may equivalently workwith U = Uω∆t,~v and matrix powers thereof.

The time trajectory effected by the rotation applied to~r results in a circle embedded within the Bloch sphere.Knowing the circle would allow to determine ~v, and thedynamics on the circle to determine ω. By constructionwe also know one point on the circle already, namely ~r(the initial condition), but we only have access to theexpectation value of an observable M for t > 0. Inthe following, we denote the “measurement direction” by~m ∈ R3, i.e., M is parametrized as M = ~m · ~σ. Algo-rithm 1 facilitates a recovery of ω and ~v using the pro-jections of the time trajectory on ~m. The main idea isto first reconstruct ω based on the time dependence, andthen the unit vector ~v.

As caveat, the solution to this problem is not unique,due to the four possible signs of the coefficients α2 andα3 in the algorithm. Fig. 3a visualizes how two rotations

can generate the same projection onto the z-axis, with ~h′

resulting from (α2, α3) → −(α2, α3). The two equationsfor α2 and α3 are shown in Fig. 3b. (In the example,α1 = −0.2051, and hence the radius of the norm con-straint circle is close to 1.) In order to decide betweenthese distinct solutions, one could perform one furthermeasurement in a different basis, or may use a-prioriknowledge about the Hamiltonian.

Algorithm 1 Reconstruction of a single-qubit Hamilto-nian (measurement direction ~m ∈ R3 with ~m ∦ ~r)

Input: Measurement averages yq = ~m · (Uωtq,~v ~r) withtq = ∆t γq for q = 0, . . . , 2dOutput: Hamiltonian parameters ω and ~v.

1: Find ω based on the dependency on ωt in Eq. (10):

ω = argminω>0

mina,b,c

2d∑q=0

|yq − a cos(ωtq − b)− c|2

2: Set yq = yq − cos(ωtq)(~m · ~r) for q = 0, . . . , 2d (subtractterm which is independent of ~v)

3: Find α1 = ~m · (~v × ~r) and κ = (~v · ~r)(~m · ~v) via a leastsquares fit:

α1, κ = argminα1,κ

2d∑q=0

|yq − sin(ωtq)α1 − (1− cos(ωtq))κ|2

4: Represent ~v with respect to the orthonormal basis{~u1 =

~r × ~m

‖~r × ~m‖ , ~u2 =~r + ~m

‖~r + ~m‖ , ~u3 =~r − ~m

‖~r − ~m‖

}:

~v = α1~u1 + α2~u2 + α3~u3,

with to-be determined coefficients α2, α3 ∈ R. Using that~u1, ~u2 and ~u3 are eigenvectors of K = 1

2(~m⊗ ~r + ~r ⊗ ~m)

with respective eigenvalues 0, λ+ = 12(1+ ~m ·~r) and λ− =

− 12(1− ~m · ~r), it follows that

κ = ~v ·K~v = λ+α22 + λ−α

23.

Together with the normalization condition

α21 + α2

2 + α23 = 1,

this leads to α22 = κ− (1− α2

1)λ− and α23 = 1− α2

1 − α22.

Decide between the four possible signs of α2, α3 using a-priori information of H or one additional measurement ina different basis.

We remark that the nonlinear optimization in the firststep of the algorithm might get trapped in a local mini-mum, which could be resolved by restarting the optimiza-tion. On the other hand, when neglecting uncertaintiesassociated with the measurements, a correct solution isindicated by a zero residual both in steps 1 and 3. Weassume that a range of realistic frequencies ω is knownbeforehand, such that the Nyquist condition (given thenon-uniform sampling points tq) holds [42].

B. Relaxation method

As straightforward approach for determining a unitarytime step matrix U which matches the measurement av-erages, one could start from the natural parametriza-tion U = e−iH (with the time step already absorbedinto H), and then optimize the Hermitian matrix H di-rectly. However, this method encounters the difficulty ofa complicated optimization landscape with many local

5

minima, in particular when using gradient descent-basedapproaches.

Here we discuss an alternative numerical approach tai-lored towards generic (dense) few-qubit Hamiltonians.As discussed in Sect. IV A, for single qubits a transfor-mation ψ′ = Uψ by a unitary matrix U ∈ SU(2) canequivalently be described by a spatial rotation in threedimensions on the Bloch sphere: ~r′ = Ur, with U ∈ SO(3)given by Rodrigues’ formula (10). An equivalent map-ping from U to U is via

Uα,β =1

2tr[σαUσβU†

](11)

for α, β = 1, 2, 3, where we have used the relation (7)together with the orthogonality relation of the Pauli ma-trices: 1

2 tr[σασβ ] = δαβ . For our purposes, the Blochpicture has the advantage that measurement averages de-pend linearly on the Bloch vector, and hence on U, e.g.,〈ψ′|σz|ψ′〉 = ~ez · ~r′ = ~ez · (Ur). In practice, we first opti-mize the entries of U to reproduce the measurement data(under the constraint that U is an orthogonal matrix),and afterwards find U related to U via Eq. (11).

Generalization to a larger number of qubits is feasiblevia tensor products of Pauli and identity matrices (inother words, Pauli strings). The analogue of (11) for nqubits, with U ∈ SU(2n), is U ∈ SO(4n − 1) with entries

Uα,β =1

2ntr[(σα1 ⊗ · · · ⊗ σαn)U(σβ1 ⊗ · · · ⊗ σβn)U†

],

(12)where α and β are now index tuples from the set{0, 1, 2, 3}⊗n\{(0, . . . , 0)}, using the convention that σ0 isthe 2×2 identity matrix. We exclude the tuple (0, . . . , 0)here since it conveys no additional information: the iden-tity matrix is always mapped to itself by unitary conju-gation.

Specifying an element of SO(4n−1) involves more freeparameters than for a unitary matrix from SU(2n) in casen ≥ 2, thus the optimization might find an orthogonalU which reproduces the measurement data, but does notoriginate from a U ∈ SU(2n) via (12). We circumventthis representability issue as follows, focusing on the caseof two qubits here. Every U ∈ SU(4) can be decomposedas [43, 44]

U =(ua ⊗ ub

)R(uc ⊗ ud

)(13)

with the “entanglement” gate

R = e−i2 (θ1 σ

1⊗σ1+θ2 σ2⊗σ2+θ3 σ

3⊗σ3), θ1, θ2, θ3 ∈ R(14)

and single-qubit unitaries ua, ub, uc, ud ∈ SU(2). Thesingle qubit gates can be handled as before, i.e., rep-resented by orthogonal rotation matrices ua, ub, uc, ud ∈SO(3). We find the Bloch representation of R via (12)(cf. [45, 46]); the parameters θ1, θ2, θ3 appear in the ma-trix entries solely as cos(θj) and sin(θj) for j = 1, 2, 3. Insummary, we express the decomposition (13) in terms ofto-be found orthogonal matrices in the Bloch picture.

After switching to the described Bloch representation,we “relax” the condition that an involved (real) matrix Uis orthogonal, by admitting any real matrix, but addingthe term

Lorth =∥∥UUT − I∥∥2

F(15)

to the overall cost function, where ‖·‖F denotes the Frobe-nius norm. As advantage, we bypass a parametrization ofU to enforce strict orthogonality and avoid local minimain the optimization.

By choosing γ = 2 in Eq. (4), the time steps tq = γq

are integers, and we can generate corresponding powersof U by defining U0 = U, Uq = U2

q−1 for q = 1, . . . , 2d,such that

e−iγqH = Uγ

q

= Uq for q = 0, . . . , 2d. (16)

In our case, the Uq’s are separate matrices, which are setin relation via an additional cost function term

Lsteps =2d∑q=1

∥∥Uq − U2q−1

∥∥2

F. (17)

For the case of two-qubit Hamiltonians, we addi-tionally substitute two-dimensional vectors (cj , sj) for(cos(θj), sin(θj)), again to avoid local minima. The con-dition c2j + s2

j = 1 translates to another penalty term inthe overall cost function:

Lθ =

3∑j=1

|c2j + s2j − 1|2. (18)

C. Partial (subsystem) measurements

An intriguing possibility of the time-delay measure-ments is the identification of the overall Hamiltonianbased on measurements restricted to a subsystem. Thisscenario will actually require the preparation of morethan one exactly known initial state. As minimal ex-ample, consider a two-qubit system, the choice betweentwo initial states, being able to select a measurement ba-sis via the gate C, and time-delayed measurements onone of the qubits while the other qubit is inaccessible, asillustrated in Fig. 4.

|ψ〉 e−iHtC

FIG. 4: Minimal example for the time-delayed partial (sub-system) measurement scenario. The goal is to reconstruct the(unknown) Hamiltonian H. We assume that one can preparetwo different initial states, and can select a measurement basisvia the gate C.

For the numerical experiment shown in Fig. 8 below,we use X-, Y - and Z-basis measurements on the top

6

qubit and minimize the mean squared error between theobserved measurement averages and the prediction basedon a general Ansatz for the (traceless) Hamiltonian, i.e.,15 real parameters.

D. Lattice system with local interactions

Finally, we consider quantum systems defined on a lat-tice, and assume an Ising-type Hamiltonian as in Eq. (1)(instead of a generic Hamiltonian) to avoid an exponen-tial growth of the number of parameters.

For the purpose of reconstructing the Hamiltonian pa-rameters, the Schrodinger time evolution can be well ap-proximated via a second order Strang splitting method[47], i.e., the Time-Evolving Block Decimation (TEBD)algorithm [48]. Interpreting the resulting layout of single-and two-site unitary operators as forming a quantum cir-cuit, the reconstruction task amounts to a variationalcircuit optimization to reproduce the reference measure-ment averages. Specifically, when considering the inter-action and local field parts of the Hamiltonian

Hint = −∑〈j,`〉

Jj,` σzjσ

z` , Hloc = −

∑j∈Λ

~hj · ~σj (19)

by themselves, the individual terms in each of the sumspairwise commute. Fig. 5 illustrates the correspondingquantum circuit for a one-dimensional lattice; the con-structing works for higher-dimensional lattices as well.

FIG. 5: One time step ∆t of the quantum dynamics basedon Strang splitting into interaction and local field terms, seeEq. (19), illustrated for a one-dimensional lattice with n = 5sites. The single qubit gates on the left and right are rotation

gates exp(−i∆t~hj · ~σj/2) for the j-th qubit, and the two-qubit gates are exp(−i∆tJj,j+1σ

zjσ

zj+1). The different shades

indicate different parameters at each site. The ordering of thetwo-qubit interaction gates among each other is not relevantsince they pairwise commute.

V. NUMERICAL EXPERIMENTS

Here we present numerical experiments for the meth-ods described in Sect. IV.

A. Exact reconstruction of a single-qubitHamiltonian

With the algorithm described in Sec. IV A we are ableto reconstruct a single-qubit Hamiltonian up to numer-ical precision. Fig. 6 shows the results obtained for theω and least squares optimizations. Following the Takensframework, we use 2d+ 1 time points, where d = 3 is thenumber of Hamiltonian parameters to be reconstructed.Specifically, we have used the time points tq = 0.3×(1.3)q

for q = 0, . . . , 6 here. With this, we are able to find ω, α1

and κ up to an error of ∼ 10−15. Thus 7 measurements,plus a further measurement in a different basis to distin-guish between the possible solutions due to symmetry,are sufficient for the reconstruction.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4t

0.1

0.2

0.3

0.4

0.5

0.6

0.7 Fitted cosineyq

(a) cosine fit to find ω

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4t

0.0

0.2

0.4

0.6

0.8Fitted functionyq

(b) function fit to find α1 and κ

FIG. 6: (a) step 1 and (b) step 3 of Algorithm 1 using Z-basismeasurements, for the time evolution shown in Fig. 3a.

B. Relaxation method

In this subsection we consider a generic single- or two-qubit Hamiltonian H. Specifically, we draw the coef-ficients of H with respect to the Pauli basis from thestandard normal (Gaussian) distribution.

We first focus on a single-qubit 2 × 2 Hamiltonian.Fig. 7a shows the result of directly fitting the vector~h ∈ R3 (see Eq. (8)), with the KL divergence betweenthe predicted and actual measurement probabilities as

7

0 2000 4000 6000 8000 10000 12000epoch

10 14

10 12

10 10

10 8

10 6

10 4

10 2

100 H rel. errorloss

(a) direct parameter fitting (single qubit)

0 10000 20000 30000 40000 50000epoch

10 14

10 12

10 10

10 8

10 6

10 4

10 2

100 U rel. errorH rel. errorloss

(b) “relaxation” method (single qubit)

0 100000 200000 300000 400000epoch

10 7

10 5

10 3

10 1

U rel. errorH rel. errorloss

(c) “relaxation” method (two qubits)

FIG. 7: (a) and (b): Single-qubit Hamiltonian reconstructionbased on four Z- and fourX-basis measurements, (a) by directfitting of the Hamiltonian parameters to the measurementdata, and (b) using the “relaxation” procedure (Sect. IV B),with the actual Hamiltonian parameters determined from U atthe end (blue line at the rightmost section of the plot). Solidlines show the median, and lighter shades are 25% quantiles.(c) “Relaxation procedure” for two-qubit Hamiltonian recon-struction, working with the Bloch picture of the representa-tion (13) to find the time step matrix U, and then determininga two-qubit Hamiltonian giving rise to this U.

cost function. This approach is compared to the “re-laxation” procedure (described in Sect. IV B) in Fig. 7b,using γ = 2 and ∆t = 0.1. The manifold for the Takensembedding has dimension d = 3 for a single qubit, andhence 2d+1 = 7 time points should be used. However, weface the difficulty that the Hamiltonian is not uniquelyspecified solely by Pauli-Z measurements according tothe discussion in Sect. IV A. For this reason, we includePauli-X measurement data as well, and reduce the num-ber of time points to 4 (to compensate for this additionalsource of data). For both versions we use the Adam opti-mizer [49]. The direct fitting method might get trappedin a local minimum and hence not be able to find theground-truth vector ~h, which leads to a large variationaround the median. This issue is ameliorated by the re-laxation method.

In Fig. 7c visualizes the loss function and relative er-rors when applying the relaxation procedure to recon-struct the unitary time step matrix and correspondingHamiltonian of a two-qubit system (d = 15 real parame-ters). Specifically, for parameter optimization we expressEq. (13) using the Bloch picture, i.e., in terms of orthog-onal rotation matrices for the single-qubit unitaries, andlikewise using the Bloch picture analogue of the entangle-ment gate (14), parametrized by two-dimensional vectors(cj , sj) for (cos(θj), sin(θj)). The condition c2j + s2

j = 1translates to another penalty term in the overall costfunction, see Eq. (18). We set ∆t = 0.05, γ = 2 and use 6time points for this experiment. The reference measure-ment data at each time point are the expectation valuesof the nine observables {I⊗σα, σα⊗I, σα⊗σα}α∈{x,y,z}.We use 54 measurement data points in total (instead of2d + 1 = 31) since we found that the additional infor-mation improves the reconstruction. As last step (afterthe main optimization), we find the Hamiltonian entriesgiving rise to the computed U via the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm.

According to Fig. 7c, the median relative errors of Uand the Hamiltonian are below 10−3, but there are stillcases where the method cannot find the reference solutionat all, even tough the loss value is small. An explanationcould be that U and H are not uniquely determined. Weleave a clarification of this point for future work.

C. Partial (subsystem) measurements

We study the scenario depicted in Fig. 4, namely per-forming measurements solely on one out of two qubits.The observables are the three Pauli gates here. For thereconstruction to work, we require that two different, pre-cisely characterized initial states can be prepared, whichwe choose at random for the numerical experiments.

The ground-truth Hamiltonian is similarly constructedat random, by drawing standard normal-distributed co-efficients of Pauli strings, which in sum form the Hamil-tonian. We use the time step ∆t = 1

5 and γ = 1.15 here,and have heuristically found 12 time-delayed measure-

8

ments (for each measurement basis) as viable to reliablyreconstruct the Hamiltonian. The loss function is themean squared error between the model prediction for themeasurement averages and the ground-truth values. Toavoid getting trapped in local minima during the numer-ical optimization, we use the best out of 10 attempts (interms of the loss function) with different random startingpoints.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9time t

0.5

0.4

0.3

0.2

0.1

0.0

0.1

0.2

(t)|(X I)| (t)(t)|(Y I)| (t)(t)|(Z I)| (t)

(a) example trajectory

0 2000 4000 6000 8000 10000epoch

10 28

10 24

10 20

10 16

10 12

10 8

10 4

100 H rel. errorloss

(b) reconstruction error and loss function

FIG. 8: Numerical reconstruction of a generic two-qubitHamiltonian based on time-delayed measurements of only oneof the qubits, cf. Fig. 4. (a) Exemplary measurement timetrajectory of the operators σx ⊗ I, σy ⊗ I and σz ⊗ I. Tworandom initial states and their respective trajectories are usedas input to the reconstruction algorithm. (b) Reconstructionerror and loss function based on 20 random realizations.

The specific observables are the three Pauli gates ap-plied to the first qubit. Fig. 8a shows exemplary measure-ment time trajectories, and Fig. 8b the reconstructionerror and loss function (median and 25% quantiles basedon 20 random realizations of the overall setup). One ob-serves that a reliable and precise reconstruction of theHamiltonian (even up to numerical rounding errors) ispossible in principle, when neglecting inaccuracies of themeasurement process.

We remark that we have used only a single initial stateand less time points for the “relaxation” method (seeFig. 7c), which can explain the seemingly larger errors

there.

D. Hamiltonian on a lattice with local interactions

We first consider the case of a Hamiltonian (1) withuniform parameters (not depending on the lattice site),i.e.,

H = −∑〈j,`〉

J σzjσz` −

∑j∈Λ

~h · ~σj (20)

with J ∈ R and ~h ∈ R3. Thus the task consists ofreconstructing 4 real numbers. The ground-truth val-

ues for the following experiment are J = 1 and ~h =(0.5,−0.8, 1.1). Λ is a 3× 4 lattice with periodic bound-ary conditions, and the initial state is a single wavefunc-tion with complex random entries (independently nor-mally distributed). We compute the time-evolved quan-tum state ψ(t) via the KrylovKit Julia package [50], anduse the squared entries of ψ(t) as reference Born mea-surement averages.

For the purpose of reconstructing J and ~h, we approx-imate a time step via a variational quantum circuit asshown in Fig. 5, where the single- and two-qubit gates

share their to-be optimized parameters J and~h. For

simplicity, we use three uniform time points ∆t, 2∆t,3∆t with ∆t = 0.2 (instead of time points tq = γq∆t),such that the quantum circuit for realizing a single timestep can be reused.

We quantify the deviation between the exact Bornmeasurement probabilities, p(t) = |ψ(t)|2, and the proba-bilities p(t) resulting from the trotterized circuit Ansatz,via the Kullback-Leibler divergence:

DKL (p(t) ‖ p(t)) =∑j

pj(t) logpj(t)

pj(t). (21)

Fig. 9a visualizes the optimization progress, i.e., the

relative errors of J and ~h, and the loss function (21).The darker curves show medians over 100 trials of ran-dom initial states, and the lighter shades 25% quantiles.The dashed horizontal line is the loss function evaluatedat the ground truth values of J and ~h; it is non-zero dueto the Strang splitting approximation of a time step. In-terestingly, the optimization arrives at an even smallerloss value starting around training epoch 70. We inter-pret this as an artefact of the Strang splitting approxi-mation – note that around this epoch, the relative error

of ~h slightly increases again. We have used the RMSPropoptimizer [51] with a learning rate of 0.005 here. In sum-mary, the reachable relative error is around 0.02, andhigher accuracy would likely require a smaller time stepto reduce the Strang splitting error.

Next, we consider a Hamiltonian (1) with disorder (andexternal field solely in x-direction):

H = −∑〈j,`〉

Jj,` σzjσ

z` −

∑j∈Λ

hj · σxj (22)

9

0 20 40 60 80 100 120epoch

10 2

10 1

100 J rel. errorh rel. errorlossref. loss

(a) uniform coefficients

0 25 50 75 100 125 150 175 200epoch

10 3

10 2

10 1

100 Jz rel. errorhx rel. errorlossref. loss

(b) random coefficients (disorder)

FIG. 9: Relative reconstruction error of the Hamiltonian pa-rameters for the 3 × 4 lattice model in (a) Eq. (20) and (b)Eq. (22), and loss function during training, when fitting aquantum circuit representation of a time step (Fig. 5) to theexact Born measurement averages.

with i.i.d. random coefficients Jj,` and hj . For the nu-merical experiment, Jj,` is uniformly distributed in theinterval [0.8, 1.2], and hj ∼ 0.5N (0, 1) (normal distribu-tion).

The results are shown in Fig. 9b. As before, we evalu-ate 100 realizations of the experiment, with a set of coef-ficients and random initial state drawn independently foreach trial. We take the maximum over the relative errorsof Jj,` for all j, ` to arrive at the relative error reportedin Fig. 9b, and likewise for hj . The “reference” loss func-tion is the KL divergence evaluated at the ground truth

coefficients. It is non-zero due to Strang splitting errors,and fluctuates due to the different random coefficients ateach trial.

VI. CONCLUSIONS AND OUTLOOK

Our work demonstrates the feasibility of using mea-surements at different time points to reconstruct the timeevolution operator. The approach presented in this workhas two main benefits: First, it requires only a single(or few) initial state(s) and reduces the number of dif-ferent kinds of measurements for the reconstruction ofthe Hamiltonian. From an experimental point of view,this would be helpful when the experimental setup makesit easier to maintain the time evolution of the systemthan to implement a variety of measurements. Second,as demonstrated in Sect. V C, the method allows for thereconstruction of a Hamiltonian even when only part ofthe system can be observed.

A natural extension of our work is QPT in the situ-ation of dissipation and noise processes, i.e., a systemgoverned by a Lindblad equation. This would pose theadditional challenge of a limited time window for extract-ing information and a larger number of free parameters.An interesting alternative approach could be a mappingto a unitary evolution with an unobserved environment[52], which would fit into the present framework.

The relaxation method in Sect. IV B can be regardedas tool to smooth the optimization landscape, but anopen question is how to guarantee convergence to thecorrect solution. This becomes particularly relevant forlarger systems and the likewise increasing number of freeparameters.

A related task for future work is a sensitivity analysiswith respect to inaccuracies and noise in the measure-ment data. A modified version of Takens theorem stillapplies in this situation [41], but it is not clear how accu-rate our algorithm can reconstruct H. A first step couldbe a gradient calculation of the measurement averageswith respect to the Hamiltonian parameters.

Acknowledgments

We thank the Munich Center for Quantum Science andTechnology for support.

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