Quantum measurement is a crucial subject in quantum computation and communication [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 79, 81, 20, 1, 2, 3, 83, 84, 85, 86, 87, 88]. 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 projective [30, 31, 32, 33, 34, 35, 36, 37] or POVM (positive-operator valued measure) measurements [80, 82, 33, 41, 42, 43, 44, 45].
Quantum measurement is required element in high-complexity quantum computations, in high-performance 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 high-precision control in the physical layer through several rounds, which results in a high-cost 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 , which produces a classical output string . The aim is then to find a global optimal output that describes the properties of the output quantum system with the highest accuracy according to quality measurement functions. An example of a high-cost 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 . Without loss of generality, the -length input quantum system of the quantum circuit is assumed to be a superposed quantum system that is fed into the circuit. Then, the -length output quantum system is measured by the measurement operator , which produces a string and, after repeating the procedure times, yields the global optimal string with success probability . Assuming that is an arbitrary quantum circuit and 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 standard measurement rounds, the achievable success probability is approximately ). Since each measurement round requires high-cost and high-precision quantum state preparations and quantum operations, the total cost to find the global optimal 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 measurement rounds, such that rounds leads to a success probability of . The dense measurement strategy is rooted in the theory of compressed sensing [54, 55, 56, 57], which allows recovering noisy signals with a high efficiency in the field of traditional communications. Dense quantum measurement utilizes an randomized measurement operator that is defined as an
-bit length vector, where
is a random variable,, , associated with the measurement of the -th quantum state of the output quantum system, while is a quantum measurement in the computational basis ; thus, if and if . As follows, the measurement in the computational basis is discarded if . Then, the measurement result is post-processed via unit that integrates algorithms to determine the global optimal string from the results of the randomized measurements.
As we prove (see Theorem 2), the number of standard measurement rounds can be reduced to dense measurement rounds for an arbitrary quantum circuit, where and , while are constants. At this number of measurement rounds, the success probability is for any practical value of . We also prove that if the output of the quantum circuit is a computational basis quantum state, then can be reduced to dense measurement rounds, where is a constant, such that for any (see Theorem 3).
The novel contributions of our manuscript are as follows:
We define a novel quantum measurement theory called dense quantum measurement.
We prove that dense measurement reduces the number of required measurement rounds to find a global optimal output.
We prove that dense measurement significantly improves the success probability of finding a global optimal output.
We provide an application scenario for quantum circuits with arbitrary unitary sequences, and for the dense measurement of computational basis quantum states in gate-model quantum computer environment.
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.
2 Related Works
The related works on quantum measurement theory, gate-model quantum computers and compressed sensing are summarized as follows.
2.1 Quantum Measurement Theory
. 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 measurement [30, 31, 32, 33, 34, 35, 36, 37], 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 measurement . In our manuscript the projective measurement with no post-processing on the measurement results is referred to as standard measurement111It is motivated by the fact, that in a gate-model 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 measurement [33, 41, 42, 43, 44, 45]. 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 operator [38, 39, 40]. The POVM measurements therefore can be expressed in terms of projective measurements (see also Neumark’s dilation theorem [46, 47, 48]).
2.2 Gate-Model Quantum Computers
, the authors studied the subject of objective function evaluation of computational problems fed into a gate-model 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 gate-model 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 one-qubit length and multi-qubit 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 gate-model quantum computers, optimization of objective function estimation, measurement optimization and optimization of post-processing in a gate-model quantum computer environment. In our particular work we are focusing on the optimization of the measurement phase.
An optimization algorithm related to gate-model quantum computer architectures is defined in 
. 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 gate-model 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 gate-model computer environment.
In , 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) [18, 19] is also proposed in the work. The work concluded that the QAOA can be implemented on near-term gate-model quantum computers for optimization problems.
An application of the QAOA algorithm to a bounded occurrence constraint problem “Max E3LIN2” can be found in . 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 gate-model quantum computer environment.
In , 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.
In , the authors analyzed the experimental implementation of the QAOA algorithm on near-term gate-model 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 non-adiabatic mechanisms.
In , the authors studied the implementation of QAOA with parallelizable gates. The work introduced a scheme to parallelize the QAOA for arbitrary all-to-all connected problem graphs in a layout of qubits. The proposed method was defined by single qubit operations and the interactions were set by pair-wise CNOT gates among nearest neighbors. As the work concluded, this structure allows for a parallelizable implementation in quantum devices with a square lattice geometry.
, the authors defined a gate-model quantum neural network. The gate-model quantum neural network describes a quantum neural network implemented on gate-model quantum computer. The work focuses on the architectural attributes of a gate-model 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 gate-model 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 gate-model quantum neural networks represent a practically implementable solution for the realization of quantum neural networks on near-term gate-model quantum computer architectures.
In , 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 -SAT defined in . The work also studied the performance of the proposed quantum algorithm and compared it with the classical algorithm.
For a review on the noisy intermediate-scale quantum (NISQ) era and its technological effects and impacts on quantum computing, see .
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 in . For a work on the complexity-theoretic foundations of quantum supremacy, see .
2.3 Compressed Sensing
In traditional information processing, compressed sensing  is a technique to reduce the sampling rate to recover a signal from fewer samples than it is stated by the Shannon-Nyquist sampling theorem (that states that the sampling rate of a continuous-time signal must be twice its highest frequency for the reconstruction) [54, 55, 56, 57]. 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) [57, 58, 59, 60, 61, 62]. Along with the sparsity, the restricted isometry property [57, 58, 62] 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 well-studied problem in the field of compressed sensing [63, 64, 65, 66, 66, 67].
A special technique within compressed sensing is the so-called “1-bit” compressed sensing [68, 69, 70], where 1-bit measurements are applied that preserve only the sign information of the measurements.
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 gate-model quantum computer environment.
3 Problem Statement
Let be the superposed input system of a quantum circuit with a quantum gate structure, formulated by quantum states, as
where is the dimension of the quantum system, is a computational basis state and is the unitary operation of , defined as a sequence of unitaries
where is the -dimensional vector of the gate parameters of the unitaries (gate parameter vector):
In (2), an -th unitary gate is evaluated as
is a generalized Pauli operator formulated by the tensor product of Pauli operators.
In a standard measurement setting, the output of is
measured by a measurement operator, which yields an output string as
The global optimal output string is an output string that yields the optimal estimation at a particular objective function fed into the quantum circuit as a maximization problem
where is the estimate yielded in an -th measurement round, , while is the output string yielded in the -th round.
Without loss of generality, after measurement rounds, the probability that the global optimal output string is determined is ; thus, can be found with the same success probability,
The problems connected to the general measurement strategy to find are the high number of repetitions and the low success probability. Consequently, the standard measurement procedure requires high-cost quantum state preparations, the application of high-cost measurement arrays and high-precision control and calibrations in the physical layer.
Problems 1-3 summarize the problems to be solved.
(System Model). Define a novel quantum measurement strategy for the significant reduction of the measurement rounds of standard measurements and for the significant improvement of the success probability in determining a global optimal output .
(General application). Define and for an arbitrary quantum circuit with . Prove the number of measurement rounds, , and the success probability, .
(Dense measurement of computational basis quantum states). Define and for an arbitrary quantum circuit with , where sets the computational basis 222Throughout the manuscript, the term “computational basis” refers to a basis , for which holds at a given , where is an input system.. Prove the number of measurement rounds, , and the success probability, .
The resolutions of Problems 1-3 are given in Theorems 1-3, respectively.
4.1 Sub-Gaussian Distributions
A random variable is sub-Gaussian, if for the probability distribution of ,
holds for , where
are sub-Gaussian parameters.
By theory, if is sub-Gaussian with
then there exists a constant depending on only such that
5.1 System Model
(Dense measurement). A structure with unitary , where the unitary sets an arbitrary computational basis for an -length input as , such that , , holds for the -norm of , where is a classical representation of , while is the actual setting of the unitaries of at and with a random measurement operator, allows the determination of the global optimal output and global optimal estimate at a particular objective function as holds, where and are constants depending on , where and is the measurement operator of the -th dense measurement round .
Proof. First, we rewrite (2) as
where is a unitary that sets a computational basis and is a unitary operation that sets the unitaries, such that
where is the identity and is the -dimensional vector of the gate parameters of . Applying the unitary on input system yields the -length quantum system ,
where the computational basis for in (17) is selected such that for the -norm of the following relation holds
is a classical representation of , is a classical representation of and . Therefore, can be an arbitrary computational basis for which (20) holds at a given (21) (For example, if is the Fourier basis, then
realizes a quantum Fourier transform).
whose state is measured by an random measurement operator, defined as an -bit length vector
where is a random variable,
associated with the measurement of the -th quantum system of in (22), and is a measurement in the computational basis .
Thus, the measurement of the -th quantum system of is defined via the following rule:
In other words, the measurement result is kept only if in (23); otherwise, the measurement result is discarded and replaced by a zero element. This results output , as
while is an -length classical vector formulated via the bits of (24) as
Thus, from (34), the output vector is evaluated as
Then, assume that the procedure repeats for rounds. The rounds of dense measurement are defined via an measurement matrix as
where is an -size random measurement vector (23) of the -th measurement round , as
where is the -th bit of defined via (24), and of the -th round is
and of the -th round is
For the rounds, define the orthogonal matrix as
and the measurement output matrix as
where is the measurement result vector (33) of the -th round.
The problem is therefore to find the optimal value of , such that the total error probability at the end of rounds
picks up a given arbitrary value that is determined via the success of the minimization (32) in the unit.
After some argumentations on the probability distribution of (42), at measurement rounds a concentration relation can be written as
for where .
where , if is selected as
The motivation for the selection of is as follows. The value of in (48) guarantees that the relation holds with probability , as it is given in (47). If is greater than (48), then , while if is lower than the value given in (48), then . As a corollary, the lowest value of to satisfy the relation with probability at least , is as given in (48). To prove (48), express via (46) as
where is subset, is a submatrix,
is the identity matrix,is the cardinality of subset and is the set of natural numbers not exceeding .
for , and . Let
where is a Hermitian matrix,
Therefore, can be expressed as a maximization
that leads to relation
Then, the union bound takes over all subsets of cardinality , yields the relation of
It can be verified that in (58) for with , the relation
it can be verified that
Thus, (59) is satisfied only if
Then, setting in (63) to
Note that it also can be shown that for in (64), there exists a finite subset of a unit ball such that is
for , such that for
By finding the values of and , the relation
can be satisfied for .
It can be proven at (54) and
for that the relation
holds. Thus, for a given and , such that ,
Then, a maximization over yields
Note that if is selected to be greater than (79), the probability is increased to .
5.2 Dense Measurement Rounds in Gate-Model Quantum Computers
5.2.1 Arbitrary Unitary Sequences
The next theorem reveals that the number of dense measurement rounds can be used to determine with an error probability , such that depends only on the properties of the unitaries, while it does not depend directly on the actual .
(Dense measurements at a quantum gate structure). For an arbitrary unitary in with , the global optimal and estimate can be determined via dense measurement rounds, with probability , where , is the -th element of the -th column of , while is a constant.
Proof. Let assume that can be decomposed as , and the following bound can be formulated for the entries of ,
where is the -th element of the -th column of , and
Let assume that the size of is , with columns , . Then let be the normalization of as
where the normalized columns form an orthonormal system, and let be the inner product of two normalized columns and , as
that can be rewritten as
where and . Therefore, in (84), the sum operator runs over the elements of the -th column of unitary , and the elements of the -th column of , respectively.
Then, at and , some argumentations on bounded orthonormal systems straightforwardly yields the boundedness condition 
where is the -th column of .
Then, for the maximal entry of , a bound can be established via the normalized columns, as
Then, by introducing a projector that selects a subset of in the rounds, the (37) measurement operator applied on a unitary can be rewritten as
where is a subset of elements selected uniform at random from all subsets of of cardinality , ,
As follows, the (43) measurement result can be rewritten as
It is required to verify that the error probability (44) at a projector in (87) is bounded by an error probability associated with the selection of rows uniformly and independently at random from .
Thus, we define set with the same cardinality as (88), such that its elements are selected independently and uniformly at random from ,
Then, let be a subset of selected uniform at random from all subsets of of cardinality , ,
For any subset , we define a failure event as
i.e., the event that the -minimization (i.e,. a basis pursuit algorithm in ) allows no to determine every from (89) on (Note that the success probability of an -minimization in to determine is independent from the normalization of the measurement operator.).
It can be verified, that for ,
and for ,
and if holds for , then
where is the distribution.
probability of event at