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An approximate diagonalization method for large scale Hamiltonians
Mohammad H. Amin,1,2Anatoly Yu. Smirnov,1Neil G. Dickson,1and Marshall Drew-Brook1
1D-Wave Systems Inc., 100-4401 Still Creek Drive, Burnaby, British Columbia, Canada V5C 6G9
2Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
An approximate diagonalization method is proposed that combines exact diagonalization and
perturbation expansion to calculate low energy eigenvalues and eigenfunctions of a Hamiltonian.
The method involves deriving an effective Hamiltonian for each eigenvalue to be calculated, using
perturbation expansion, and extracting the eigenvalue from the diagonalization of the effective
Hamiltonian. The size of the effective Hamiltonian can be significantly smaller than that of the
original Hamiltonian, hence the diagonalization can be done much faster. We compare the results of
our method with those obtained using exact diagonalization and quantum Monte Carlo calculation
for random problem instances with up to 128 qubits.
I.INTRODUCTION
Diagonalization of large Hermitian matrices is a diffi-
cult problem in linear algebra, with applications in a va-
riety of disciplines. In quantum mechanics, for example,
the energy levels of a quantum system are obtained by
diagonalization of the system’s Hamiltonian. Knowing
those energy levels is necessary for describing the behav-
ior of the quantum system, e.g., the evolution of a quan-
tum computer consisting of N quantum bits (qubits).
Calculating the exact energy spectrum of a Hamiltonian
with usual numerical computation methods is possible
only for up to N ≈ 20 qubits. For larger systems, the
size of the Hilbert space (2N) becomes too large for the
current level of available computer memory and speed.
Although it is extremely difficult to calculate the exact
spectrum of a multi-qubit system for large N, there are
ways to calculate an approximate spectrum that shows
important features of the exact spectrum reliably over
a range of parameters. These methods include Density
Matrix Renormalization [1–4], Lanczos [5], and Quantum
Monte Carlo calculations [6–9]. Perturbation theory is
also an approximate method, that is applicable when the
system’s Hamiltonian is close to a simpler Hamiltonian,
called the unperturbed Hamiltonian, for which the eigen-
values and eigenfunctions are known or easy to calculate.
For example, the unperturbed Hamiltonian can be diag-
onal in some known basis and the perturbation Hamilto-
nian can have small off diagonal elements in such a basis.
One can perform perturbation expansion in powers of a
small parameter, characterizing the off-diagonal terms of
the Hamiltonian, to find approximate solutions for the
eigenvalues and eigenfunctions of the total Hamiltonian.
A true perturbation expansion provides a Taylor ex-
pansion of the energy levels in powers of a small param-
eter.However, such expansion can become extremely
complicated when there are energy level degeneracies in
the spectrum of the unperturbed Hamiltonian. More-
over, the perturbation expansion can quickly break down
if the energy separations of the unperturbed states are
small or when there are anticrossings between the eigen-
states in the spectrum.Here, we combine perturba-
tion expansion with exact diagonalization techniques to
achieve an effective method for approximate diagonaliza-
tion. The idea is to separate a subspace, e.g., low en-
ergy states of the unperturbed Hamiltonian, from other
(high energy) states in the Hilbert space. If the unper-
turbed Hamiltonian is diagonal, then it might be easy
to find its lowest energy states using the structure of
the problem.Starting from the original Hamiltonian,
we derive an effective Hamiltonian in the subspace using
perturbation expansion. Perturbation theory brings into
consideration, in the expansion of each term of the ef-
fective Hamiltonian, the relevant states that are outside
the subspace. If the unperturbed states in the subspace
are non-degenerate, then there won’t be a unique effec-
tive Hamiltonian that can provide perturbed eigenvalues
after the diagonalization. Instead, there will be an effec-
tive Hamiltonian for each non-degenerate unperturbed
state. Since the calculation involves exact diagonaliza-
tion of these effective Hamiltonians, the final results are
not true Taylor expansions in powers of the small param-
eter, as in usual perturbation expansion. Yet, perturba-
tion plays an important role in the derivation of these
effective Hamiltonians.
II.THE FORMALISM
To derive the effective Hamiltonians, we adopt the pro-
jection operator approach to perturbation theory as dis-
cussed by Yao and Shi [10]. Consider the Hamiltonian
H = H0+ V , in which H0is the unperturbed Hamilto-
nian and V is the perturbation Hamiltonian. The aim of
the perturbation theory is to find the eigenvalues Enand
eigenvectors |n? of H, with
(H0+ V )|n? = En|n?,
assuming that the eigenvalues E(0)
|n(0)? of H0are known.
Consider subspace S in the Hilbert space of H0, con-
sisting of NSvectors |k(0)?. The subspace S can include
both degenerate and non-degenerate eigenstates of H0.
We introduce projection operators
?
(1)
n
and eigenvectors
P =
k∈S
|k(0)??k(0)|,
¯P = 1 − P.(2)
arXiv:1202.2817v1 [quant-ph] 13 Feb 2012
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Since H0, P, and¯P are diagonal in |k(0)? basis, we have
[P,H0] = [¯P,H0] = 0, (3)
Let |k? denote an eigenstate of H that we are trying to
calculate. We write |k? = |k?P+|k?¯ P, where |k?P≡ P|k?
and |k?¯ P≡¯P|k?. We multiply both sides of (1), written
for the eigenstate |k?, by P and¯P, respectively, to get
E(0)
k|k?P+ PV (|k?P+ |k?¯ P) = Ek|k?P,
H0|k?¯ P+¯PV (|k?P+ |k?¯ P) = Ek|k?¯ P,(4)
which can be rewritten as
(Ek− E(0)
(Ek− H0−¯PV¯P)|k?¯ P=¯PV P|k?P.
Here, we have used P2= P and¯P2=¯P. Solving the
second equation for |k?¯ P, we get
|k?¯ P= (Ek− H0−¯PV¯P)−1¯PV P|k?P
k
− PV P)|k?P= PV¯P|k?¯ P,
(5)
(6)
Substituting (6) into the first equation in (5), we find
? H(k)|k?P= Ek|k?P,(7)
where? H is an NS×NSmatrix defined by
? H(k) ≡ E(0)
with I being the NS×NSunity matrix.
So far, Eq. (7) is exact. What it means is that Ekis
an eigenvalue and |k?P is an eigenvector of? H(k) in S.
Ekand |k?P, but no information about other eigenvalues
of H is obtained. Notice that Ekappears in both (8) and
(7), therefore it has to be calculated self-consistently. As
we shall see, perturbation theory can help to calculate
Ek, order by order.
If? H is independent of k, then all the perturbed eigen-
agonalization. For a k-dependent? H, on the other hand,
ing, while all other eigenvalues do not correspond to the
correct eigenenergies of the spectrum of H. In that case,
to calculate each Ek, one has to calculate its correspond-
ing? H(k), diagonalize it, and select the right eigenvalue
this way will not be orthogonal to each other. This indeed
should be the case because only the original eigenstates
|k? in the full Hilbert space are supposed to be orthog-
onal to each other, hence the projected states |k?P may
not be orthogonal.
As mentioned earlier,? H(k) is a function of Ek, which
kI + PV P
+ PV¯P(Ek− H0−¯PV¯P)−1¯PV P,(8)
Therefore, formally by diagonalizing? H(k), one can find
values and eigenvectors in S can be found in a single di-
only one of the eigenvalues, i.e., Ek, has physical mean-
that corresponds to Ek. Note that the states |k?P found
itself has to be calculated via diagonalization of? H(k).
The calculation becomes tractable using perturbation ex-
pansion. Consider the expansion
(Ek− H0−¯PV¯P)−1= (E(0)
∞
?
where, δEk= Ek− E(0)
? H(k) ≡ E(0)
j=0
k
− H0−¯PV¯P + δEk)−1
?j
=
j=0
?
(E(0)
k−H0)−1(¯PV¯P−δEk)
(E(0)
k−H0)−1,
k. Substituting into (8), we get
kI + PV P + PV¯P ×
(9)
∞
?
?
(E(0)
k−H0)−1(¯PV¯P−δEk)
?j
(E(0)
k−H0)−1¯PV P.
Writing δEk= E(1)
denote the order of perturbation, one can calculate? H
operator¯P(E(0)
k
− H0)−1¯P is not singular and is given
by
k
+ E(2)
k
+ ..., where the superscripts
order by order. Because of the projection operator¯P, the
¯P(E(0)
k
− H0)−1¯P =
?
n/ ∈S
|n(0)??n(0)|
E(0)
k
− E(0)
n
.(10)
We now derive analytical formulas for all the elements
of? H(k) up to the forth order perturbation. Defining
perturbed states in S, and assuming that V has only
off-diagonal elements in the chosen basis so that E(1)
?k(0)|V |k(0)? = 0, we find
? H(0)
? H(2)
? H(3)
? H(4)
where, E(0)
n
and
Vαβ≡?α(0)|V |β(0)?, where |α(0)? and |β(0)? denote un-
k
=
αβ(k) = E(0)
? H(1)
kδαβ,
αβ(k) = Vαβ,
αβ(k) =
?
?
?
n/ ∈S
VαnVnβ
E(0)
kn
VαnVnmVmβ
E(0)
,(11)
αβ(k) =
n,m/ ∈S
knE(0)
VαnVnmVmpVpβ
E(0)
km
,
αβ(k) =
n,m,p/ ∈S
knE(0)
kmE(0)
kp
− E(2)
k
?
n/ ∈S
VαnVnβ
[E(0)
kn]2,
kn=E(0)
k−E(0)
E(2)
k
=? H(2)
kk(k) =
?
n/ ∈S
VknVnk
E(0)
kn
.(12)
Notice that for each added order of perturbation, a factor
of the form Vnm/E(0)
The small parameter of the expansion, therefore, should
be k-dependent and have the form:
kmis added to the expansion terms.
λk∼ min
n,m/ ∈S[Vnm/E(0)
km].(13)
In our numerical calculations, we found the best agree-
ment with exact diagonalization when expanding the di-
agonal elements to the forth order of perturbation, but
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the off-diagonal elements to the second order.
can be understood if one considers only two levels, i.e.,
NS= 2. By diagonalizing the 2×2 reduced Hamiltonian
corresponding to the two levels, if the diagonal elements
are not the same, a second order off-diagonal element
contributes a forth order term to the final eigenvalues, as
it gets squared. Therefore to be consistent in the order of
perturbation, one should expand the off-diagonal terms
to the second order.
Notice that all the energy differences in the denomi-
nators are of the form E(0)
kmand therefore depend on the
unperturbed energy E(0)
k
of state |k(0)?. As a result the
calculated? H is k dependent, unless all the states in S
rection will still have k-dependence through the second
term in the last equation of (11)). As we mentioned be-
fore, one cannot obtain all eigenstates by a single diag-
onalization of? H(k). Instead, one has to calculate? H(k)
task then is to select, among all the eigenvalues of? H(k),
the perturbation of |k(0)?. If the perturbed levels do not
cross each other, then Ekwill be the k-th eigenvalue after
the diagonalization. In cases when the perturbed states
do cross each other, the situation becomes more compli-
cated. One can use the overlap of the new eigenfunctions
with the old ones to identify which two correspond to
each other.
The accuracy of the calculations depends on the small
parameter of the perturbation expansion, λk. To have
an estimate of λk using (13), let Emin represent the
lowest energy level outside the subspace S, therefore
|Ekm|m/ ∈S≥ Emin−E(0)
for the small parameter: λk≤ max(Vnm)/(Emin−E(0)
Perturbation expansion, thus becomes more accurate for
the lowest energy states for which E(0)
the accuracy of the calculations increases by increasing
Emin, i.e., increasing NS. In principle, there is no limit to
the accuracy and therefore no fixed radius of convergence
as in the usual perturbation theory. By taking NS→ N,
one can achieve 100% accuracy and an unlimited radius
of convergence. In practice, however, NSis limited by the
limitation of the computation time and available mem-
ory. By keeping NS small, the diagonalization can be
done very quickly, but at the price of less accurate re-
sults. Quite naturally, for NS< N, small features of the
spectrum that depend on the contribution of the higher
energy states, beyond S and the states included pertur-
batively, cannot be reproduced.
This
are degenerate (even in that case, the forth order cor-
for each unperturbed eigenstate |k(0)?. The important
the right eigenvalue Ekthat corresponds to state |k?, i.e.,
k. This provides an upper bound
k).
k
is smallest. Also,
III.COMPARISON WITH EXACT
DIAGONALIZATION AND QUANTUM
MONTE-CARLO SIMULATION
To test our approximate diagonalization method, we
study several Hamiltonians with different sizes and com-
00.20.4 0.60.81
0
1
2
3
4
5
6
7
8
s
Energies (GHz)
ε
∆
FIG. 1: Hamiltonian parameters ∆ and E as a function of
normalized time s.
pare our results with those of the exact diagonalization
and quantum Monte Carlo simulation. The Hamiltonian
we consider is an Ising Hamiltonian in a transverse field
of the form
?
HP =
i
H(s) = −1
2∆(s)
?
i
σx
i+1
2E(s)HP,(14)
hiσz
i+
?
i<j
Jijσz
iσz
j,(15)
where, s ∈ [0,1], hi and Jij are dimensionless param-
eters that can be adjusted, and ∆(s) and E(s) are en-
ergy scales plotted in Fig. 1. Hamiltonian (14) was stud-
ied in Ref. 14 to investigate the scaling performance of
an adiabatic quantum computation (AQC) [11] proces-
sor based on realistic Hamiltonian parameters. In that
case, s = t/tf represents normalized time, where tf is
the total evolution time. It is known [11] that in AQC,
the minimum energy gap between the lowest two energy
levels during the evolution determines the time of the
computation. Therefore it is important to diagonalize
the Hamiltonian (14) to calculate the minimum energy
gap. Details of the AQC processor considered in this
study are described in other publications [12, 13]. Here,
we only focus on the diagonalization of the Hamiltonians.
In Ref. 14, a number of random Ising instances were
generated and the size of the minimum gap in the spec-
trum of their Hamiltonians was calculated using QMC
simulation.With QMC simulation, one can calculate
the energy gap between the lowest two energy levels us-
ing the method discussed in [9, 14]. The instances used in
Ref. 14 were generated by choosing hiuniform randomly
from the set {-1/3,1/3} and a structured set of nonzero
Jij values to be either -1, or uniform randomly from {-
1/3,1/3}. The connectivity of the graph considered was
motivated by a realistic quantum annealing processor as
described in [13, 14]. Here, we use the same set of prob-
lems and compare our results with the QMC results of
Ref. 14.
First, we need to find the unperturbed eigenstates and
eigenvalues of Hpfor the instances studied. Since Hpis
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0.40.50.6 0.70.8 0.91
0
2
4
6
8
s
E−E0 (GHz)
N=8
0.40.50.6 0.70.80.91
1
2
3
4
5
6
7
8
s
E−E0 (GHz)
N=16
FIG. 2: Spectra of 8-qubit (a) and 16-qubit (b) random Ising
Hamiltonians with transverse fields, relative to the ground
state energy E0. Solid thin lines represent the approximate
diagonalization results and dashed lines represent exact diag-
onalization results. Circles show the results of the quantum
Monte Carlo simulations.
already diagonal, it is only needed to determine the states
with the lowest energy to form S. For that to be feasi-
ble up to 128 qubits, we use the structure of the graph of
nonzero couplings between the qubits. We use a dynamic
programming method that is a variation on the bucket
elimination algorithm [15]. We select an order in which to
“eliminate” the qubits, i.e. to solve for the optimal value
of a qubit conditional on the values of all qubits coupled
to it that have not yet been eliminated. After eliminat-
ing all qubits, the lowest energy state is simply retrieved
by tracing back from the optimal value of the qubit that
was eliminated last. To find the NSlow energy states, we
keep track of the energy increase from choosing the sub-
optimal value of each qubit being eliminated, and while
tracing back through the elimination, the lowest energy
NS partial states encountered so far are kept, instead
of just the optimal partial state. With this approach,
the ability to find these unperturbed states is primar-
ily limited by the largest number of qubits that need to
be simultaneously considered during the elimination (the
treewidth of the graph), instead of the total number of
qubits. For the instances considered, the treewidth was
up to 16.
0.40.50.6 0.70.80.91
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1
2
3
4
5
6
s
gap (GHz)
N=32
0.40.50.60.70.80.91
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6
1
2
3
4
5
s
gap (GHz)
N=48
0.40.50.60.70.80.91
0
6
1
2
3
4
5
s
gap (GHz)
N=72
0.4 0.50.60.70.80.91
0
6
1
2
3
4
5
s
gap (GHz)
N=96
0.4 0.50.6 0.70.80.91
0
1
2
3
4
5
s
gap (GHz)
N=128
FIG. 3: Approximate energy gap (solid lines) of example ran-
dom Ising instances compared with results from Monte Carlo
simulations (circles). From top to bottom N=32,48,72,96, and
128.
We choose the unperturbed and perturbation Hamil-
tonians in the following way:
H0=1
2E(s)HP,V = −1
2∆(s)
?
i
σx
i.(16)
The small parameter in the perturbation expansion,
therefore, is proportional to ∆(s)/E(s). In our calcu-
lations we kept NS∼ 2000−6000 in the subspace S. We
choose NSin such a way that all the degenerate states in
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the topmost energy level are included in the subspace S.
A simplifying observation, for the calculation of (11), is
that V only contains terms with operators of the form σx
which flips the state of qubit i in the σz
V , therefore, only one bit flip from the original state is
allowed. Consequently, in the calculation of the matrix
elements of effective Hamiltonians? H(k), using (11), only
participate in the sums. This significantly restricts the
states |m?, |n?, or |p? that are summed over in (11). The
states outside S were found by flipping qubits away from
states in S, as required by the perturbation expansion.
We have examined many instances and compared the
calculated approximate eigenvalues with the QMC simu-
lation results of Ref. [14] and also, for small size instances,
with the exact diagonalization results. Here, however, we
only report a few sample instances. Figure 2(a) and (b)
show the calculated spectra for of 8-qubit and 16-qubit
sample problems, respectively. Due to the small num-
ber of qubits, the exact diagonalization was possible for
these instances. The figures show excellent agreement
between the exact (dashed lines) and approximate (solid
lines) diagonalization methods over a wide range of the
normalized time s. Even complicated details of the exact
spectra are nicely reproduced by the approximate diago-
nalization. As expected, some of the higher energy curves
deviate from the exact diagonalization values at small s,
where the perturbation expansion starts to fail. By in-
creasing NS, one can increase the validity range of the
calculation at the expense of a longer computation time.
The QMC results (symbols) for the above two instances
are also plotted in the same figures. As can be seen, QMC
agrees very well with the two other methods for s<
For larger values of s, QMC simulation fails to give re-
liable results due to the small tunneling amplitudes. As
we shall see below, the same pattern continues for larger
scale problems.
For problems with N > 16, it is not feasible to perform
exact diagonalization as the size of the Hamiltonian be-
comes exponentially large. As a consequence, we only
compare our results with those calculated using QMC
simulations. Figure 3 shows the calculated gap between
the lowest two energy levels, for N from 32 to 128. For
most instances the approximate diagonalization results
i
ibasis. For each
states with at most two bit flips from states |α? and |β?
∼0.7.
agree with QMC calculations for 0.4<
fore, QMC fails to give the correct spectral gap for large
s. Also, for the NS values chosen, the perturbation ex-
pansion becomes less reliable for s<
accuracy can always be enhanced by increasing NS. In-
teresting examples are N = 48 and 128 for which there
are anticrossings in the spectrum. The position of the
anticrossing and the shape of the energy levels close to
it are more or less consistent between the two methods
of calculation. The size of the minimum gap at the an-
ticrossing point, however, can depend on states outside
the subspace S that are not included in the perturbation
calculation. Therefore, the minimum gap size cannot be
reliably predicted unless a very large number of states
are included in S, or alternatively, the perturbation is
expanded to high orders.
∼s<
∼0.7. As be-
∼0.4, although the
IV.CONCLUSIONS
We have developed an approximate diagonalization
method for calculating low energy eigenvalues and eigen-
functions of a large scale Hamiltonian. The method is
based on derivation of a series of effective Hamiltonians,
in a subspace consisting of low energy states of an unper-
turbed Hamiltonian, using perturbation expansion. For
each eigenvalue to be calculated, an effective Hamilto-
nian is calculated and diagonalized separately. We have
applied our method to find the energy eigenvalues of ran-
dom Ising Hamiltonians in a transverse field. Our results
agree very well with the exact diagonalization for 8 and
16 qubit Hamiltonians, and with quantum Monte Carlo
simulations for up to 128 qubits. The approximate di-
agonalization method, however, is extremely faster than
both of the above methods.
Acknowledgment
We thank Elena Smirnova and Elena Tolkacheva for
critically reading the manuscript and providing valuable
comments.
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