Dense Quantum Measurement Theory
Dense Quantum Measurement theory
Laszlo Gyongyosi 0 1 2
sandor Imre 1
0 School of Electronics and Computer Science, University of Southampton , Southampton, SO17 1BJ , UK
1 Department of Networked Systems and Services, Budapest University ofTechnology and Economics , Budapest, H1117 , Hungary
2 MTABME Information Systems Research Group, Hungarian Academy of Sciences , Budapest, H1051 , Hungary
Published: xx xx xxxx Quantum measurement is a fundamental cornerstone of experimental quantum computations. the main issues in current quantum measurement strategies are the high number of measurement rounds to determine a global optimal measurement output and the low success probability of finding a global optimal measurement output. each measurement round requires preparing the quantum system and applying quantum operations and measurements with highprecision control in the physical layer. these issues result in extremely highcost measurements with a low probability of success at the end of the measurement rounds. Here, we define a novel measurement for quantum computations called dense quantum measurement. The dense measurement strategy aims at fixing the main drawbacks of standard quantum measurements by achieving a significant reduction in the number of necessary measurement rounds and by radically improving the success probabilities of finding global optimal outputs. We provide application scenarios for quantum circuits with arbitrary unitary sequences, and prove that dense measurement theory provides an experimentally implementable solution for gatemodel quantum computer architectures.

Quantum measurement is a crucial subject in quantum computation and communication1?31. The aim of
quantum measurement is to extract valuable and useable information from the measured quantum system. The
measurement operator connects the quantum world and our traditional, classical world. While the input of the
measurement can be a superposed or entangled quantum system, the output of the measurement is classical
information (i.e., bitstrings). Quantum measurements can be performed in different ways, for example via
projective32?39 or POVM (positiveoperator valued measure) measurements35,40?46.
Quantum measurement is required element in highcomplexity quantum computations, in highperformance
quantum information processing and in quantum computer architectures. The main issues of current quantum
measurement strategies are the high number of measurement rounds and the probability of successfully finding
a global optimal measurement output. The necessity of a high number of measurement rounds requires preparing
the input quantum system and applying quantum operations with highprecision control in the physical layer
through several rounds, which results in a highcost procedure overall that is not tractable in any experimental
setting. The repetition of a measurement round therefore requires in each round the careful preparation of a
quantum register of quantum states that are then fed into a quantum circuit that realizes an arbitrary unitary
sequence. In each round, the output of the quantum circuit is measured by a measurement array M, which
produces a classical output string z. The aim is then to find a global optimal output z* that describes the properties of
the output quantum system with the highest accuracy according to quality measurement functions. An example
of a highcost application of standard measurement is measuring the output of a quantum circuit applied to
realize quantum computations where the quantum circuit is set to perform a unitary operation U. Without loss of
generality, the nlength input quantum system X of the quantum circuit is assumed to be a superposed quantum
system that is fed into the circuit. Then, the nlength output quantum system Y = U X is measured by the
measurement operator M, which produces a string z and, after repeating the procedure R0 times, yields the global
optimal string z* with success probability PrR0(z ?). Assuming that U is an arbitrary quantum circuit and M is a
standard measurement, the measurement procedure requires high repetition numbers, while the success
probability remains low (An example is the application of standard quantum measurements in quantum computers,
where for R0 ? 100 standard measurement rounds, the achievable success probability is approximately
PrR0(z ?) ? 0.0113). Since each measurement round requires highcost and highprecision quantum state
preparations and quantum operations, the total cost to find the global optimal z* is very high in a practical setting. To
avoid the issues of a high number of measurement rounds and the low success probability of quantum
measurements, a novel measurement is essential for quantum computations.
Here, we define a novel measurement for quantum computations called dense quantum measurement. The
dense measurement strategy aims at fixing the drawbacks of standard quantum measurements by achieving a
radical reduction in the number of necessary measurement rounds and by significantly improving the success
probabilities of finding global optimal outputs (see Theorem 1 for the system model). Dense quantum
measurement requires only R ? R0 measurement rounds, such that R rounds leads to a success probability of
Pr(z ?) ? PrR0(z ?). The dense measurement strategy is rooted in the theory of compressed sensing47?50, which
allows recovering noisy signals with a high efficiency in the field of traditional communications. Dense quantum
measurement utilizes an Mr randomized measurement operator that is defined as an nbit length vector
Mr = (b1M , ?, bnMB)T, where bi is a random variable, bi ? {0, 1}, Pr(0) = Pr(1) = 0.5, associated with the
B
measurement of the ith quantum state of the output quantum system, while MB is a quantum measurement in the
computational basis B; thus, biMB = 0 if bi = 0 and biMB = MB if bi = 1. As follows, the MB measurement in the
computational basis is discarded if bi = 0. Then, the measurement result is postprocessed via unit ? that
integrates algorithms to determine the global optimal string z* from the results of the randomized measurements.
As we prove (see Theorem 2), the number R0 of standard measurement rounds can be reduced to
R = ?Z2K log4(n) dense measurement rounds for an arbitrary quantum circuit, where K ? L0(S) and K ? n,
while ? > 0, Z > 0 are constants. At this number of measurement rounds, the success probability is
Pr(z ?) = 1 ? n?log 3(n) ? 1 for any practical value of n. We also prove that if the output of the quantum circuit is
rounds, where ? > 0 is a constant, such that Pr(z ?) = 1 ? 2 exp(?R) ? 1, for any R (see Theorem 3).
The novel contributions of our manuscript are as follows:
a computational basis quantum state, then R0 can be reduced to R = ?(?K log( 1K0n )) dense measurement
1. We define a novel quantum measurement theory called dense quantum measurement.
2. We prove that dense measurement reduces the number of required measurement rounds to find a global
optimal output.
3. We prove that dense measurement significantly improves the success probability of finding a global
optimal output.
4. We provide an application scenario for quantum circuits with arbitrary unitary sequences, and for
the dense measurement of computational basis quantum states in gatemodel quantum computer
environment.
5. We reveal that the primary advantages of dense quantum measurement theory are the significantly lower
measurement rounds and significantly higher success probabilities.
This manuscript is organized as follows. In Section 2, the related works are summarized. In Section 3, the
problem statement is given. In Section 4, preliminaries are summarized. Section 5 proposes the theorems and
proofs. Section 6 provides a performance evaluation. Finally, Section 7 concludes the paper.?Supplemental
information is included in the?Appendix.
Related Works
The related works on quantum measurement theory, gatemodel quantum computers and compressed sensing
are summarized as follows.
Quantum Measurement theory. Quantum measurement has a fundamental role in quantum mechanics
with several different theoretical interpretations32?44. The measurement of a quantum system collapses of the
quantum system into an eigenstate of the operator corresponding to the measurement. The measurement of a
quantum system produces a measurement result, the expected values of measurement are associated with a
particular probability distribution.
In quantum mechanics several different measurement techniques exist. In a projective measurement32?39, the
measurement of the quantum system is mathematically interpreted by projectors that project any initial quantum
state onto one of the basis states. The projective measurement is also known as von Neumann measurement32. In
our manuscript the projective measurement with no postprocessing on the measurement results is referred to
as standard measurement (It is motivated by the fact, that in a gatemodel quantum computer environment the
output quantum system is measured with respect to a particular computational basis).
The von Neumann measurements are a special case of a more general measurement, the POVM
measurement35,40?44. Without loss of generality, the POVM is a generalized measurement that can be interpreted as a von
Neumann measurement that utilizes an additional quantum system (called ancilla). The POVM measurement
is mathematically described by a set of positive operators such that their sum is the identity operator51?53. The
POVM measurements therefore can be expressed in terms of projective measurements (see also Neumark?s
dilation theorem54?56).
Another subject connected to quantum measurement theory is quantumstate discrimination57?61 that covers
the distinguishability of quantum states, and the problem of differentiation between nonorthogonal quantum
states.
GateModel Quantum Computers. The theoretical background of the gatemodel quantum computer
environment utilized in our manuscript can be found in12 and13.
In13, the authors studied the subject of objective function evaluation of computational problems fed into a
gatemodel quantum computer environment. The work focuses on a qubit architectures with a fixed hardware
structure in the physical layout. In the system model of a gatemodel quantum computer, the quantum computer
is modeled as a sequence of unitary operators (quantum gates). The quantum gates are associated with a
particular control parameter called the gate parameter. The quantum gates can process onequbit length and multiqubit
length quantum systems. The input quantum system (particularly a superposed quantum system) of the
quantum circuit is transformed via a sequence of unitaries controlled via the gate parameters, and the output qubits
are measured by a measurement array. The measurement in the model is realized by a projective measurement
applied on a qubits that outputs a logical bit with value zero or one for each measured qubit. The result of the
measurement is therefore a classical bitstring. The output bitstring is processed further to estimate the objective
function of the quantum computer. The work also induces and opens several important optimization questions,
such as the optimization of quantum circuits of gatemodel quantum computers, optimization of objective
function estimation, measurement optimization and optimization of postprocessing in a gatemodel quantum
computer environment. In our particular work we are focusing on the optimization of the measurement phase.
An optimization algorithm related to gatemodel quantum computer architectures is defined in12. The
optimization algorithm is called ?Quantum Approximate Optimization Algorithm? (QAOA). The aim of the algorithm
is to output approximate solutions for combinatorial optimization problems fed into the quantum computer. The
algorithm is implementable via gatemodel quantum computers such that the depth of the quantum circuit grows
linearly with a particular control parameter. The work also proposed the performance of the algorithm at the
utilization of different gate parameter values for the unitaries of the gatemodel computer environment.
In62, the authors studied some attributes of the QAOA algorithm. The authors showed that the output
distribution provided by QAOA cannot be efficiently simulated on any classical device. A comparison with the ?Quantum
Adiabatic Algorithm? (QADI)63,64 is also proposed in the work. The work concluded that the QAOA can be
implemented on nearterm gatemodel quantum computers for optimization problems.
An application of the QAOA algorithm to a bounded occurrence constraint problem ?Max E3LIN2? can be
found in15. In the analyzed problem, the input is a set of linear equations each of which has three boolean
variables, and each equation outputs whether the sum of the variables is 0 or is 1 in a mod 2 representation. The work is
aimed to demonstrate the capabilities of the QAOA algorithm in a gatemodel quantum computer environment.
In65, the authors studied the objective function value distributions of the QAOA algorithm. The work
concluded, at some particular setting and conditions the objective function values could become concentrated. A
conclusion of the work, the number of running sequences of the quantum computer can be reduced.
In66, the authors analyzed the experimental implementation of the QAOA algorithm on nearterm gatemodel
quantum devices. The work also defined an optimization method for the QAOA, and studied the performance of
QAOA. As the authors found, the QAOA can learn via optimization to utilize nonadiabatic mechanisms.
In67, the authors studied the implementation of QAOA with parallelizable gates. The work introduced a
scheme to parallelize the QAOA for arbitrary alltoall connected problem graphs in a layout of qubits. The
proposed method was defined by single qubit operations and the interactions were set by pairwise CNOT gates
among nearest neighbors. As the work concluded, this structure allows for a parallelizable implementation in
quantum devices with a square lattice geometry.
In14, the authors defined a gatemodel quantum neural network. The gatemodel quantum neural network
describes a quantum neural network implemented on gatemodel quantum computer. The work focuses on the
architectural attributes of a gatemodel quantum neural network, and studies the training methods. A particular
problem studied in the work is the classification of classical data sets which consist of bitstrings with binary labels.
In the architectural model of a gatemodel quantum neural network, the weights are represented by the gate
parameters of the unitaries of the network, and the training method acts these gate parameters. As the authors
stated, the gatemodel quantum neural networks represent a practically implementable solution for the
realization of quantum neural networks on nearterm gatemodel quantum computer architectures.
In68, the authors defined a quantum algorithm that is realized via a quantum Markov process. The analyzed
process of the work was a quantum version of a classical probabilistic algorithm for kSAT defined in69. The work
also studied the performance of the proposed quantum algorithm and compared it with the classical algorithm.
For a review on the noisy intermediatescale quantum (NISQ) era and its technological effects and impacts on
quantum computing, see1.
The subject of quantum computational supremacy (tasks and problems that quantum computers can solve but
are beyond the capability of any classical computer) and its practical implications are studied in2. For a work on
the complexitytheoretic foundations of quantum supremacy, see3.
A comprehensive survey on quantum channels can be found in23, while for a survey on quantum computing
technology, see70.
Compressed sensing. In traditional information processing, compressed sensing47 is a technique to reduce
the sampling rate to recover a signal from fewer samples than it is stated by the ShannonNyquist sampling
theorem (that states that the sampling rate of a continuoustime signal must be twice its highest frequency for
the reconstruction)47?50. In the framework of compressed sensing, the signal reconstruction process exploits the
sparsity of signals (in the context of compressed sensing, a signal is called sparse if most of its components are
zero)50,71?75. Along with the sparsity, the restricted isometry property50,71,75 is also an important concept of
compressed sensing, since, without loss of generality, this property makes it possible to yield unique outputs from the
measurements of the sparse inputs. The restricted isometry property is also a wellstudied problem in the field of
compressed sensing76?80.
A special technique within compressed sensing is the socalled ?1bit? compressed sensing81?83, where 1bit
measurements are applied that preserve only the sign information of the measurements.
The application of compressed sensing covers the fields of traditional signal processing, image processing and
several different fields of computational mathematics84?91.
The dense quantum measurement theory proposed in our manuscript also utilizes the fundamental concepts
of compressed sensing. However, in our framework the primary aims are the reduction of the measurement
rounds required to determine a global optimal output at arbitrary unitaries, and the boosting of the success
probability of finding a global optimal output at a particular measurement round. The results are illustrated through a
gatemodel quantum computer environment.
problem statement
Let X be the superposed input system of a quantum circuit with a QG quantum gate structure, formulated by n
quantum states, as
X =
1
dn
? z ,
z
?
? = (?1, ?, ?L)T .
Ui (?i) = exp (?i?iP),
Y
?
= U ( ? ) X
z = M Y .
C (z ?) = max C (zm),
? m
PrR0C (z ?) = PrR0 (z ?).
where P is a generalized Pauli operator formulated by the tensor product of Pauli operators {?X, ?Y, ?Z}.
In a standard measurement setting, the Y output of QG is
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
measured by a M measurement operator, which yields an output string z as
The global optimal output string z* is an output string that yields the optimal estimation C(z ?) at a particular
objective function C fed into the quantum circuit as a maximization problem
where C(zm) is the estimate yielded in an mth measurement round, m = 1, ?, R0, while zm is the output string
yielded in the mth round.
Without loss of generality, after R0 measurement rounds, the probability that the global optimal output string
z* is determined is PrR0(z ?); thus, C(z ?) can be found with the same success probability,
The problems connected to the general measurement strategy to find z* are the high number of R0 repetitions
and the low PrR0(z ?) success probability. Consequently, the standard measurement procedure requires highcost
quantum state preparations, the application of highcost measurement arrays and highprecision control and
calibrations in the physical layer.
Problems 1?3 summarize the problems to be solved.
Problem 1 (System Model). Define a novel quantum measurement strategy for the significant reduction of the R0
measurement rounds of standard measurements and for the significant improvement of the PrR0(z ?) success
probability in determining a global optimal output z*.
?
Problem 2 (General application). Define R and Pr(z ?) for an arbitrary quantum circuit withU( ? ). Prove the
number R of measurement rounds, R ? R0, and the Pr(z ?) success probability, Pr(z ?) ? PrR0(z ?).
Problem 3 (Dense measurement of computational basis quantum states). Define R and Pr(z ?) for an arbitrary
quan?
tum circuit with U( ? ) = UB, where UB sets the computational basis B (Throughout the manuscript, the term
?computational basis? refers to a basis B, for which L0(S) ? K holds at a given S = BX, where X is an input system). Prove
the number R of measurement rounds, R ? R0, and the Pr(z ?) success probability, Pr(z ?) ? PrR0(z ?).
The resolutions of Problems 1?3 are given in Theorems 1?3, respectively.
A random variable X is subGaussian, if for the probability distribution of X,
preliminaries
subGaussian Distributions.
holds for ? ? > 0, where
are subGaussian parameters.
By theory, if X is subGaussian with
Pr (X ? ?) ? C1e?C2?2
C1, C2 > 0
? (X) = 0,
? (exp(?X) ? exp (c??2))
C1 = 2,
C2 =
C1, C2 > 0
Pr ( Mj,k ? ?) ? C1e?C2?2
then there exists a constant c* depending on only C1, C2 such that
for ? ? ? R.
If (12) holds, then (11) is satisfied such that the C1 subGaussian parameter of X is
and C2 is as
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
An M ? N random matrix M is a subGaussian random matrix, if
for ? ? > 0, where Mj,k is the (j, k)th element of M, j ? [M], k ? [N], where
are subGaussian parameters.
Methods
system Model. Theorem 1 (Dense measurement). A QG structure with unitary U( ??) = UBU( ??? ), where the
unitary sets an arbitrary computational basis B for an nlength input X as UB X = S , such that L0(S) ? K ,
?
K ? n, holds for the L0norm of S, where S is a classical representation of S , while U( ?? ) is the actual setting of the
unitaries of QG at S and with a Mr random measurement operator, allows the determination of the global optimal
output z* and global optimal estimate C(z ?) at a particular objective function C as ? = Pr(?K ? ?) holds, where ?K
?
and ? are constants depending onQ = MU( ?? ), where ? = (Mr(1), ?, Mr(R)) and Mr(m) is the measurement
operator of the mth dense measurement round m = 1, ?, R.
Proof. First, we rewrite (2) as
? ?
U( ? ) = UBU( ?? ),
?
where UB is a unitary that sets a computational basis B andU( ?? ) is a unitary operation that sets the unitaries, such
that
UBU( ?? ) (UBU( ??? ))? = UBU( ?? )UB?(U( ??? ))?
? ?
? ?
= UBU( ?? ) (U( ?? ))?UB?
= UBIUB?
= UBUB? = I,
S = UB X ,
L0(S) = S 0 ? K,
? ?
where I is the identity and ?? is the Ldimensional vector of the gate parameters ofU( ?? ). Applying the unitary UB
on input system X yields the nlength quantum system S = s1, ?, sn ,
where the computational basis B for UB in (17) is selected such that for the L0norm of S the following relation
holds
where
is a classical representation of S , X is a classical representation of X and K ? n. Therefore, B can be an arbitrary
computational basis for which (20) holds at a given (21) (For example, if B is the Fourier basis, then UB realizes a
quantum Fourier transform).
The output of QG at (17) and (19) is therefore written as
where M?r is
and
and ? is
while ?C is an nlength classical vector formulated via the bi bits of (24) as
whose state is measured by an Mr random measurement operator, defined as an nbit length vector
where bi is a random variable,
Mr = (b1MB, ?, bnMB)T ,
bi = ????0, with Pr(0) = 0.5
???1, with Pr(1) = 0.5
,
associated with the measurement of the ith quantum system gi of G in (22), and MB is a measurement in the
computational basis B.
Thus, the measurement of the ith quantum system gi of G is defined via the following rule:
In other words, the measurement result Mr( gi ) is kept only if bi = 1 in (23); otherwise, the measurement result
is discarded and replaced by a zero element. This results output yi, as
This measurement strategy defines Mr (23) as a random Bernoulli vector47?50. Then, the nbit length output Y, is as
?
U( ? ) X
S = BX
?
= UBU ( ?? ) X
?
= U ( ?? ) (UB X )
?
= U ( ?? ) S
= G
= g1, ?, gn ,
biMB = ????0,
???MB, if bi = 1
if bi = 0,
.
y = ????0,
i ???MB( gi ) if bi = 1
if bi = 0,
.
Y =
=
=
Mr ( G )
?
MrU ( ?? ) S
M?r S
= ?C?
= ??CS,
?
M?r = MrU( ?? )
?C = (b1, ?, bn)T ,
?
??C = ?CU( ?? ),
?
? = U( ?? )S.
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
As follows, applying Mr (23) on G (22) is equivalent to applying M?r (28) on the computational basis state S
(19).
As (27) is determined via (23), the goal is to determine C(z) at a particular objective function C via a
postprocessing ?.
First, from Y (27), the computational basis vector S can be recovered as S? via ?, as a minimization50,
such that
where L1 is the L1norm. The ? unit utilizes a basis pursuit algorithm47?50 for the L1minimization in (32). Then,
?
using (32), ? is defined as
Thus, from (34), the output vector z is evaluated as
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
S? = arg min L1(S)
S
Y = ??CS
? ?
? = U( ?? )S?.
z = B?1?(Y )
= B?1(??)
? ?
= U( ? )X ,
C(z) = C(B?1?(Y )).
? = (Mr(1), ?, Mr(R)),
Mr(m) = (b1(m)MB, ?, bn(m)MB)T ,
?
M?r(m) = Mr(m)U( ?? ),
?
??C(m) = ? (m)U( ?? ),
C
? (m) = (b1(m), ?, bn(m))T .
C
Pr(z ? z ?) = ?
?
where ?(Y ) is the postprocessing (32) applied on Y, B?1 is the inverse basis transformation and X is a classical
representation of X . As follows, from (35), the C(z) estimate yields
Then, assume that the procedure repeats for R rounds. The R rounds of dense measurement are defined via an
n ? R measurement matrix ? as
where Mr(m) is an nsize random measurement vector (23) of the mth measurement round m = 1, ?, R, as
where bi(m) is the ith bit of Mr(m) defined via (24), and M?r(m) of the mth round is
and ??C(m) of the mth round is
where
For the R rounds, define the n ? R orthogonal matrix? as
and the measurement output matrix YR as
?
Q = MU( ?? ) = (M?r(1), ?, M?r(R)),
Y R = ? S = (Y (1), ?, Y (R)),
where Y(m) is the measurement result vector (33) of the mth round.
The problem is therefore to find the optimal value of R, such that the total error probability at the end of R
rounds
picks up a given arbitrary value ? that is determined via the success of the L1 minimization (32) in the ? unit.
After some argumentations on the probability distribution of? (42), at R measurement rounds a concentration
relation can be written as
Pr((L2(QS?))2 ? ?2(S?) ? ?(?2(S?)))
= Pr((L2(MG?))2 ? ?2(G?) ? ?(?2(G?)))
? 2 exp(?c?2R),
where c is a constant depending on the subGaussian parameters C1, C2 > 0 (see Section 4.1) of the subGaussian
matrix? (42), L2 is the Euclidean norm and ?2 is the ?2norm of a quantum system, ?2( ? ) = ?x ?(x) 2 = 1,
where ?(x) 2 = Pr(x) and ? Pr(x)dx = ? ?(x) 2 dx = 1, while ? is ? ? (0, 1).
By theory, the Kth restricted isometry constant47?50 ?K = ?K(?) of matrix? is the smallest ? ? 0 such that
(1 ? ?)?2( S ) ? (L2(? S ))2 ? (1 + ?)?2( S ),
for ? S where L0(S) ? K .
?
Then, for a given ?, the restricted isometry constant47?50 ?K of Q = MU( ?? ) satisfies relation ?K < ? with
probability
The motivation for the selection of R is as follows. The value of R in (48) guarantees that the relation ?K < ? holds
with probability1 ? ?, as it is given in (47). If R is greater than (48), then Pr(?K < ?) > 1 ? ?, while if R is lower
than the value given in (48), then Pr(?K < ?) < 1 ? ?. As a corollary, the lowest value of R to satisfy the relation
?K < ? with probability at least1 ? ?, is as given in (48). To prove (48), express ?K via (46) as
where ? is subset,?? is a submatrix, I is the identity matrix, ? is the cardinality of subset ? and[n] = {1, ?, n}
is the set of natural numbers not exceeding n.
The formula of (50) is equivalent to (46), since (46) can be rewritten as
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
where ? ? (0, 1), if R is selected as
Therefore, L2(H) can be expressed as a maximization
where
and
then
for ? ? ? [n], ? ? K and S? ? S. Let
where H is a Hermitian matrix,
Pr(?K < ?) = 1 ? ?
R = A ?12 ?????K?????9 + 2 log???? Kn ?????????? + 2 log?????2???? 1? ????????????????,
A = 2 .
3c
?K =
sup
??[n], ? =K
L2(????? ? I ),
(L2(?? S? ))2 ? ?2( S? ) ? ??2( S? )
Y? = ?? S? ,
Z ? = H S? ,
H = ????? ? I,
(L2(Y?))2 ? ?2( S? ) =
? S?, S?
=
Y?, Y?
Z ?, S?
??[n], ? =K L2(H) ? ?.
max
that leads to relation
Then, the union bound takes over all( n ) subsets ? ? [n] of cardinality K, yields the relation of
K
Pr(?K ? ?) ?
? Pr(L2(H) ? ?)
sup??[n], ? =K
? 2( n )(1 + 2 K
K ? ) exp(?c?2(1 ? 2? )2R)
? 2(eKn )K (1 + 2 K
? ) exp(?c?2(1 ? 2? )2R),
where we used that for integers m ? k > 0, ( mk )k ? ( mk) ? (ekm )k by theory47?50.
It can be verified that in (58) for ? ? [n] with ? = K , the relation
Note that it also can be shown that for ? ? (0, 0.5) in (
64
), there exists a finite subset ? of a unit ball
?? = {X, suppX ? ?, ?2( X ) ? 1} such that ? is
holds, if
since for
it can be verified that
Thus, (59) is satisfied only if
Then, setting? in (
63
) to
so that
and
yields (60)50.
and
(56)
(57)
(58)
(59)
(60)
(
61
)
(
62
)
(
63
)
(
64
)
(
65
)
(
66
)
(
67
)
R =
c(1 ? 12? )2?2 ?????log????1 + ? 2 ?????K + log?????2???? 1? ????????????????.
? =
log????1 + ? 2 ????? (1 ?12? )2 ? 134 ,
and
where
and
Thus,
holds. Thus, for a given z and x ? ?, such that L2(z ? x) ? ? ?
Then, a maximization over ? z ? ?? yields
V, z
=
?
W, x + D, z ? x
W, x +
D, z ? x
< ? + L2(H) ?2( z + x ) ?2( z ? x )
? ? + 2L2(H)?
V = H z ,
D = H z + x .
L2(H) < ? + 2L2(H)?.
L2(H) ?
?
1 ? 2?
.
As follows, there exists (
61
) such that L2(H) < ? holds, and combining it with (
70
) verifies the relation of (
62
).
To conclude the results, setting ? in (58) with equality in (
64
) leads to ?K < ? with probability
Pr(?K < ?) = 1 ? ?, as the R value of measurement rounds is
= 32c ?12 (K(9 + 2 log(Kn )) + 2 log(2( ?1 ))).
(
68
)
(
69
)
(
70
)
(
71
)
(
72
)
(
73
)
(
74
)
(
75
)
(
76
)
(
77
)
(
78
)
(
79
)
?
Note that if R is selected to be greater than (
79
), the probability is increased to Pr(?K < ?) > 1 ? ?.
Dense Measurement Rounds in GateModel Quantum Computers. Arbitrary Unitary
Sequences. The next theorem reveals that the number R of dense measurement rounds can be used to determine
z* with an error probability ? = n?log 3(n), such that R depends only on the properties of the unitaries, while it does
not depend directly on the actual ?. ?
Theorem 2 (Dense measurements at a U( ? ) quantum gate structure). For an arbitrary unitary UB in
? ?
U( ? ) = UBU( ?? ) with L0(S) ? K, the global optimal z* and estimateC(z ?) can be determined via R = ?Z2K log4(n)
dense measurement rounds, with probability Pr(z ?) = Pr(C(z ?)) = 1 ? n?log 3(n), where Z ? n max U( ??q,k) ,
? ? k,q?[n]
U( ?q,k) is the qth element of the kth column of U( ? ), while ? > 0 is a constant.
? ?
Proof. Let assume that U( ? ) can be decomposed as UBU( ?? ), and the following bound can be formulated for
?
the entries ofU( ? ),
As follows, the YR (43) measurement result can be rewritten as
?
max U( ?q,k) ?
k,q?[n]
Z ,
n
?
U( ?q,k) =
? ?
U( ?q,k) (U( ?q,k))? .
vk =
n uk ,
?kl =
n k
1 v , 1 v
n l
= uk, ul ,
? ?
whereU( ?q,k) is the qth element of the kth column of U( ? ), and
?
Let assume that the size of U( ? ) is n ? n, with columns uk, k = 1, ?, n. Then let vk be the normalization of
uk as
where the normalized columns form an orthonormal system, and let ?kl be the inner product of two normalized
columns vk and vl, as
that can be rewritten as
?kl = 1 ?n n U( ??q,k) n U ?( ??q,l) = 1 ?n n uk,q n ul?,q,
n q=1 n q=1 (
84
)
? ?
where ui,j = U( ?j,i) a?nd vi,j = n U( ?j,i). Therefore, in (
84
), the sum operator runs over the n elements of the kth
column of unitaryU( ? ), and the n elements of the lth column ofU ?( ??), respectively.
?
Then, at UB and U( ?? ), some argumentations on bounded orthonormal systems straightforwardly yields the
boundedness condition50
Z ? max
k,q?[n]
bq, u?k ,
where bq is the qth column of UB. ?
Then, for the maximal entry ofU( ?q,k), a bound can be established via the normalized columns, as
Z ? km,q?a[xn] vk,q
max
k,q?[n]
n uk,q
=
=
n km,q?a[xn] uk,q =
?
n max U( ?q,k) .
k,q?[n]
As follows, the bounds in (
85
) and (
86
) are equivalent to (
80
). ?
Then, by introducing a projector ?QR that selects a subset of U( ? ) in the R rounds, the ? (37) measurement
?
operator applied on a unitaryU( ? ) can be rewritten as
where QR ? [n] is a subset of R elements selected uniform at random from all subsets of [n] of cardinality R,
QR = R,
?
M = PQR(U( ? )),
QR = {q1, ?, qR}.
(
80
)
(
81
)
(
82
)
(
83
)
(
85
)
(
86
)
(
87
)
(
88
)
It is required to verify that the ? error probability (44) at a projector ?Q in (
87
) is bounded by an ?? error
probability associated with the selection of rows uniformly and independently at random fromU( ??)50.
Thus, we define setQ?R ? [n] with the same cardinality as (
88
), such that its elements are selected
independently and uniformly at random from [n], Q?R = R
Then, let Qk ? [n] be a subset of k ? R selected uniform at random from all subsets of [n] of cardinality k,
Qk = k,
Q?R = {q?1, ?, q?R}.
Qk = {q1, ?, qk}.
(
89
)
(
90
)
(
91
)
(92)
(93)
(94)
(95)
(96)
(97)
(98)
(99)
(100)
(101)
i.e., the event that the L1minimization (i.e,. a basis pursuit algorithm in ?) allows no to determine every S from
(
89
) on Q (Note that the success probability of an L1minimization in ? to determine S is independent from the
normalization of the measurement operator).
It can be verified, that forQ ? Q?,
and for k ? R,
and if Q?R = k holds for k ? R, then
where ?(?) is the distribution.
Therefore, the
probability of event ?(Q?R) at Q?R = k is as
?(Q?) ? ?(Q),
Pr(?(QR)) = ? ? Pr(?(Qk)) = ?k,
?(Q?R) = ?(Qk),
Pr(?(Q?R)) = ??
?? =
thus the error probability ? is bounded by ??.
As follows, using projector ?Q in (
87
), an L1minimization (basis pursuit)47?50 in the ? postprocessing phase
allows to determine the global optimal z* from (
89
), with probability
as
holds, where ? > 0, and ? is evaluated via (44) as
The value determined for R in (99) is based on the following fact. It can be shown50, that at a particular ? > 0
and Z ? 1, there exists a constant a, a ? (0, 1), such that for a given a, the relation
holds with probability (98), if
where
is the Kth restricted isometry constant of 1 ?, that is the smallest a ? 0 such that
R
while ? is as given by (42); holds for ? S, with L0(S) ? K . Since for (99), ? > 0 holds and Z ? 1 is satisfied via
(
86
), the result in (99) is straightforwardly follows from in (102) at
Thus, at (99) dense measurement rounds the global optimal output z*
is yielded with probability
?
where ?? is the optimal ? determined via ?. Therefore the C(z*) global optimal estimate of a particular objective
function C can be determined with probability Pr(C(z ?)) = Pr(z ?) as
where C(zm) is the estimate yielded in an mth round, m = 1, ?, R, zm is the recovered output vector in the mth
round, which concludes the proof. ?
?
The steps of the dense measurement for an arbitrary U( ? ) are summarized in Procedure 1.
?K = ?K( 1R ?)
Dense Measurements of Computational Basis Quantum States. The next theorem reveals that the number of
?
dense measurement rounds can be reduced if the unitaries of the quantum circuit are set asU( ? ) = UB, i.e., if the
output of the quantum circuit is a computational basis state S = UB X .
? ?
Theorem 3 (Number of dense measurement rounds at U( ? ) = UB). At U( ? ) = UB, the optimal z* and C(z*) can
be determined via R(?) = c1K log(10n/K ) + c2 log(2/?) dense measurement rounds, where ? ? (0, 1) is the error
probability of z*, while c1 > 0 and c2 > 0 are constants, that yields R = ?(K log( 1K0n )), with ? = 2 exp (?R).
Proof. In this setting, the unitaries of the QG quantum gate structure are set such that
?
U( ? ) = UB
holds, therefore the output of QG at an nlength input X is
? ?
Figure 2. (a) The Pr(C(z*)) success probabilities forU( ? ) = UBU( ?? ), in function of the dense measurement
rounds R, R = ?Z2K log4(n), at n = 1000, and K = 2, 5, 10, 20, ?K=2 = 2.47 ? 10?4, ?K=5 = 1.36 ? 10?4,
?K?=10 = 8.46 ?? 10?5, ?K=20 = 6.17 ? 10?5, and Z = n. (b) The Pr(C(z*)) success probabilities for
U( ? ) = UBU( ?? ), in function of R at K = 10, for n = 101, 102, 103, 104, 105, 106.
UB X = S ,
Y R = ? S ,
Y (m) = Mr(m) S = ?C(m)S,
i.e., it outputs an nlength computational basis quantum state S , with relation L0(S) ? K , K ? n.
The R measurements are performed according to the n ? R measurement matrix ? as defined in (37).
?
Since U( ? ) is set as given in (109), the measurement results of the R rounds formulate R dimensional output
Y R = (Y (1), ?, Y (R)) as
where the output of the mth round is
where Mr(m) is as given in (38), while ?C(m) is as in (41).
(110)
(111)
(112)
It further can be verified that ? is a subGaussian random matrix, thus there exists a constant ? > 0
depending only on the C1, C2 subGaussian parameters (see Section 4.1) of ? such that for a given ?, the restricted
isometry constant47?50 ?K = ?K(?) of ?, satisfies relation ?K < ? with probability
that concludes the proof. ?
where ?K = ?K(?) is the Kth restricted isometry constant of ?, which is the smallest ? ? 0 such that
(1 ? ?)?2( S ) ? (L2(? S ))2 ? (1 + ?)?2( S ),
for ? S, with L0(S) ? K . Note, that at ? = 2 exp ( ? ?2R2? ), (114) picks up the value of R = 2? ?12 (K log ( 1K0n )).
Then, some argumentation on the L1minimization based recovery via basis pursuit in the ? unit, yields a
condition for the 2Kth restricted isometry constant, ?2K of ? as
The condition in (116) allows to determine anyS? in the ? postprocessing as a unique solution of
such that for every S there exists a unique solution of (117).
Thus, in an mth measurement round, output vector zm is evaluated as
where Y(m) is given in (112).
Since (116) puts a strict bound on ?, it allows us to rewrite (114) at a particular
?? + c2 log??? 2 ??
R(?) = c1K log????1K0n ??? ??,
?? ? ??
where c1, c2 > 0 are constants depend only on the C1, C2 subGaussian parameters (see Section 4.1) of ?.
Therefore, the global optimal z* can be determined via R rounds via ?, without loss of generality as
where ? > 0 is a constant, that yields C(z*) via (108). Thus, at ? = 1, the success probability is
R = ????K log????1K0n ??????
?
? ????,
?
Pr(z ?) = Pr(C(z ?))
= 1 ? 2 exp( ? R) = 1 ? ?,
if
as
subject to
with success probability
where ?CR is as
Pr(?K < ?) = 1 ? ?,
The steps of the dense measurement for an computational basis quantum states are summarized in Procedure 2.
?
Procedure 2. Dense measurements atU( ? ) = U .
B
ethics statement.
This work did not involve any active collection of human data.
PrR0(C(z ?)) = Pr?R0(z ?) ? 0.01).
Performance Evaluation
In this section, we analyze the Pr(C(z ?)) = Pr(z ?) success probabilities of finding the global optimal output z*,
and global optimal estimate C(z*) in function of R, for an arbitrary objective function of the quantum computer
? ?
with a QG quantum circuit, and arbitrary objective function C. First the U( ? ) = UBU( ?? ) setting is discussed,
?
then theU( ? ) = UB situation is proposed.
? ?
In Fig.?2, a dense measurement at theU( ? ) = UBU( ?? ) case is depicted. In this case, R is evaluated as given in
(99). In Fig.?2(a) the length of the measured quantum system is fixed to n = 1000, while K varies between 5 and
20. In Fig.?2(b), the value of K is fixed to K = 10, while n varies between n = 101 and n = 106. For a comparison
the results of R0 standard measurements13 are also depicted in both figures with dashed gray lines (R0 = 100,
In Fig.?3, aU( ? ) = UB situation is depicted. In this case, a computational basis quantum state is outputted by
the QG structure, and R is evaluated as given in (124). In Fig.?3(a) the length of the measured quantum system is
fixed to n = 1000, while K varies between 5 and 20. In Fig.?3(b), the value of K is fixed to K = 10, while n varies
between n = 101 and n = 106.
Conclusions
Here, we defined a novel measurement technique called dense measurement for quantum computation. Dense
measurement utilizes a random measurement strategy and a postprocessing unit to eliminate the main
drawbacks of standard measurement techniques. The dense measurement method provides two fundamental results.
First, it significantly increases the success probability of finding a global optimal measurement result. Second,
it radically reduces the number of measurement rounds required to determine a global optimal measurement
result. We demonstrated the results through an application of dense measurements with quantum circuits that
realize arbitrary unitary operations. We proved the results of dense measurement theory for the measurement of
arbitrary quantum states and for the measurement of computational basis quantum states in gatemodel quantum
computer environment.
Data Availability
This work does not have any experimental data.
1 7
Acknowledgements
This work was partially supported by the National Research Development and Innovation Office of Hungary
(Project No. 20171.2.1NKP201700001), by the Hungarian Scientific Research Fund  OTKA K112125 and in
part by the BME Artificial Intelligence FIKP grant of EMMI (BME FIKPMI/SC).
Author Contributions
L.GY. designed the protocol and wrote the manuscript. L.GY. and S.I. analyzed the results. All authors reviewed
the manuscript.
Additional Information
Supplementary information accompanies this paper at https://doi.org/10.1038/s41598019432502.
Competing Interests: The authors declare no competing interests.
Publisher?s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and
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