Experimental Data from a Quantum Computer Verifies the Generalized Pauli Exclusion Principle

"What are the consequences ... that Fermi particles cannot get into the same state ..."R. P. Feynman wrote of the Pauli exclusion principle,"In fact, almost all the peculiarities of the material world hinge on this wonderful fact."In 1972 Borland and Dennis showed that there exist powerful constraints beyond the Pauli exclusion principle on the orbital occupations of Fermi particles, providing important restrictions on quantum correlation and entanglement. Here we use computations on quantum computers to experimentally verify the existence of these additional constraints. Quantum many-fermion states are randomly prepared on the quantum computer and tested for constraint violations. Measurements show no violation and confirm the generalized Pauli exclusion principle with an error of one part in one quintillion.


Introduction
While performing calculations with classical computers at IBM, Borland and Dennis discovered something unexpected and surprising about the electronic structure of atoms and molecules [1].
In 1926 Pauli had observed that no more than a single electron can occupy a given one-electron quantum state known as a spin orbital [2]. Formally, the Pauli exclusion principle implies that the spin-orbital occupations are rigorously bounded by zero and one. In their calculations Borland and Dennis discovered, however, that even in three-electron atoms and molecules there are additional constraints beyond the well-known exclusion principle. In 2006 Klyachko (and in 2008 with Altunbulak) presented a systematic mathematical procedure for generating these constraints for potentially arbitrary numbers of electrons and orbitals [3,4]. These inequalities, which have become known as generalized Pauli constraints [5,6,7,8], provide new insights into the electron structure of many-electron atoms and molecules [7,9,10,11,12], the limitations of entanglement as a resource for quantum control [13,14], as well as the fundamental distinctions between open and closed quantum systems [15].
In this work we use quantum states prepared on a quantum computer to provide experimental verification of the generalized Pauli constraints. Quantum computers differ from classical computers in that quantum states can be prepared on the quantum computer. Algorithms which utilize these quantum states promise a large computational advantage over known classical algorithms for solving critically important problems such as integer factorization, eigenvalues estimation, and fermionic simulation [16,17,18,19]. Rapid advances in quantum hardware and hybrid classical-quantum algorithms have led to multi-qubit experimental implementations [20,21,22]. We first randomly prepare the quantum state of a 3-electron system and second measure the occupations of the natural orbitals. The natural orbitals are the eigenfunctions of the 1-electron reduced density matrix (1-RDM) that is defined by integrating the many-electron density matrix over the coordinates of all electrons except one in which Ψ(123) is the 3-electron wave function. The two-step process is repeated many times on the quantum computer to explore all possible physically realizable orbital occupations.
Because of the generalized Pauli constraints a large convex set of orbital occupations should be experimentally forbidden. In contrast to the classical computations of Borland and Dennis, we are not representing the quantum states by matrices but rather preparing quantum states directly, which allows us to measure the orbital occupations experimentally.

Measurement of 1-RDM Eigenvalues
Pauli observed in 1926 that for quantum systems of fermion particles such as electrons the occupation n i of each spin orbital must obey the following inequalities: known as the Pauli exclusion principle or Pauli constraints. In 1963 Coleman proved mathematically that these constraints plus a normalization constraint in which the occupation numbers sum to N are necessary and sufficient for the occupation numbers n i to represent at least one ensemble state of N electrons [23]. Borland and Dennis in 1972, however, discovered that there exist additional conditions on the occupation numbers for the representation of at least one pure state of N electrons, which are presently known as the generalized Pauli constraints [1]. A pure state is a quantum state that is describable by a single wave function. Borland and Dennis found the following generalized Pauli constraints for three electrons in six orbitals: where n 1 + n 6 = 1 (4) where n i are the natural-orbital occupations ordered from largest to smallest.
To test the generalized Pauli constraints on a quantum computer, we prepare an initial pure state |Ψ 0 (123) of 3 fermions in 6 orbitals and perform arbitrary unitary transformationsÛ i of the initial state to generate a set of random pure states |Ψ i (123) of 3 fermions in 6 orbitals We measure the matrix elements of the 1-RDM of each state |Ψ i (123) generated on the quan-  The system of 3 fermions in 6 orbitals can be expressed as a system of 3 qubits, allowing the above procedure to be simplified. Because the occupations of each pair, {n 1 , n 6 }, {n 2 , n 5 }, and {n 3 , n 4 }, must sum to one, the three electron systems has a one-to-one mapping to a system of three qubits. The occupation numbers n 4 , n 5 , and n 6 , which are eigenvalues of the 1-RDM, can be viewed as the eigenvalues p 1 , p 2 , and p 3 of the 1-qubit reduced density matrix of a threequbits system. Hence, for the three-qubits system the generalized Pauli constraints in Eqs. 3-6 can be written as the single inequality where the parameters α, β, and γ in the Pauli rotation matrices R y (α) are chosen randomly and the C j i are controlled NOT (CNOT) gates. The rotation matrices are applied to the control qubit of the ensuing CNOT gate. It is known that any 3-qubit state can be prepared from the non-interacting state by a unitary transformation built from only 3 CNOT gates plus universal single-qubit gates [25]. The above transformation generates states that span the most general entanglement class for the system and whose 1-qubit RDMs cover all possible real 1qubit occupation numbers [26,27,14]. Computations were also performed with a slightly more general transformation, discussed in the Methods and Supplementary Figure 1, albeit without a significant change in the results. Because of the mapping between the 3-qubit and the 3-fermion-in-6-orbitals system, the measured occupations p 1 , p 2 , and p 3 are equivalent to the natural occupations (eigenvalues) n 4 , n 5 , and n 6 of the 1-RDM. For the remainder of this work we primarily discuss the results in terms of the 3-fermion system.

Verification of Generalized Pauli Constraints
The scatter plot of the measured 1-RDM natural occupations is shown in Fig. 2  result is statistically robust [14]. The Pauli polytope is twice the size of the generalized Pauli polytope. Consequently, without further restrictions beyond the ordinary Pauli constraints the probability of a randomly prepared state being in the yellow region would be one out of two.
The probability of n random states being on the yellow side, therefore, would be 1/2 n . With n being approximately 60, we observe that the probability of measuring all 60 points within the yellow region would be 1/2 60 or a highly improbable one in one quintillion. Hence, we have verified the generalized Pauli exclusion principle by quantum computer to a high degree of confidence.
Despite the restrictions on the pure-state observables from the generalized Pauli constraints, the set of realizable 1-RDMs exhibits all limits of quantum behavior including the mean-field limit as well as the strong-electron-correlation limit including phenomena like superconductivity. Figure 3 shows examples of chemical systems which widely vary in the degree of correlation present. As observed in previous work, for 3-electron-in-6-orbital systems the ground-state  system [10]. Such conditions can potentially be used to enhance the direct variational calculation of the 2-RDM [28], which is applicable to treating strongly correlated atomic and molecular quantum systems at a polynomial-scaling computational cost [29,30,31].

Methods
We include details on the quantum algorithm used in the article and its variants, the method for generating the parameters, the quantum tomography of the one-electron reduced density matrix, and relevant details on the experimental quantum device used.

Quantum Algorithms
Two algorithms were used in this work. Both utilize 3 CNOT gates, which allow for the whole set of 3-qubit occupation numbers to be spanned with suitable single qubit rotations. A single CNOT can form a biseparable system, while two CNOT gates can reach the GHZ state and spread a far range of the polytope, but three CNOT gates are required to saturate the Higuchi inequality [24,25,26,27]. Complex unitary transformations can be used, increasing the set of all possible quantum states, but these transformations span the same set of occupations as real rotations.
The first algorithm takes a minimalistic approach, and spans the polytope by parameterizing only three rotations. The algorithm (applied right to left) is as follows: where R y,i refers to a qubit rotation around the y-axis of the Bloch Sphere onto the i th qubit (later we drop the y, as we only use R y rotations), defined as: .
C j i is the standard CNOT gate with i control and j target qubits. The control qubit is rotated prior to a CNOT transformation. The sequence of transformations produce a wave function of the form: where α, β, γ, and δ are all functions of θ 1 , θ 2 , and θ 3 , and the wave function has no diagonal elements in the 1-RDM.
The second algorithm provides a over-redundant representation of the system, including some local degrees of freedom [27], and involves rotations on both the target and control qubit prior to the rotation, and is as follows: Note that the CNOT identity implies that by performing certain single-qubit rotations on both qubits, we somewhat eliminate the dependence of the qubit orderings with a generic rotation. Of course the wave function is of a more general form, involving all 8 qubit states, and so the off-diagonal elements need to be determined.

Generating Parameters
The parameters for each run were obtained by classically simulating the circuit over a range of parameters (from 0 • to 45 • in 0.1 • intervals), calculating the theoretical eigenvalues after applying a given algorithm, and then either: 1) adding the point to a set if a point was further from all other points by a set distance, or; 2) discarding the point. Thus for the first algorithm we sampled 90 million points, and for the second, 8 quadrillion. The minimum distance for inclusion was 0.075. Only 62 and 59 points were selected for the first and second respectively.
More points could have been used, but this would not have provided additional clarity, due to the scale of the shifts from errors. In general, unbiased random sampling did not yield 'graphically' uniform sets, as there is a tendency to cluster in certain areas of the polytope, and would lead to lots of runs being only in a particular region. Thus, the above method was used.

Quantum Tomography of the 1-RDM
The quantum tomography of the 3-electrons-in-6-orbitals system is simplified by its mapping to a system of 3 qubits. We derive the unitary transformations required to perform quantum tomography of the 1-RDM for a completely general many-electron state and outline the simplified application of this tomography for the 3-electrons-in-6-orbitals system.

Consider the unitary transformation in terms of the rotation angle
whereα whereâ † i andâ i are second-quantized operators that create and annihilate an electron in orbital i. We can express this unitary transformation U in the following closed form: whereβ Using the Baker-Campbell-Hausdorff expansion, we evaluate the unitary transformation of the projection operatorM for measuring the occupation of the i th orbital If we set the angle φ to π/4 and take the expectation value with respect to the quantum state |Ψ , we obtain By an analogous procedure we obtain from which we can extract the imaginary off-diagonal elements of the 1-RDM.
Because of Eqs. (4-6) of the Borland-Dennis constraints the 6 natural orbitals of the 3electrons-in-6-orbitals system can be paired to form 3 qubits where each qubit is a two-level system sharing an electron. We can restrict the one-body unitary transformation in Eq. (16) to indices i and j representing orbitals of the same qubit because other choices violate the restriction of one electron per qubit. Hence, the one-body fermionic unitary transformation can be represented as a unitary transformation of a single qubit where Γ is a global phase from the antisymmetry of electrons. This mapping to qubits can be viewed as a member of the family of compact mappings [32]. Practically, this transformation is implemented in this study for φ = π/4 as a product of the Hadamard gate H and a Pauli-Z gate Z. If Γ = 1 and φ = π/4, then U = HZ, and if Γ = 1 and φ = π/4, then U = ZH.

Quantum Computation
In this work we used the IBM Quantum Experience devices (ibmqx4 and ibmqx2), available online, in particular the 5-transmon quantum computing device [33]. These cloud accessible quantum devices are fixed-frequency transmon qubits with co-planer waveguide resonators [33,34]. Experimental calibration for these devices is included below, and connectivity is specified there. Additionally, results are included in Supplementary Tables 1-8.
For our work, we tested varying number of measurements, and found that no significant decrease in the error of a run occurred by using more than 2048 measurements, and so the For 2048 and 1024 iterations, this yields a standard error of 0.007, and 0.010, respectively, and the 95% confidence interval is then ±0.014, and ±0.020, respectively. For other points similar values were seen, typically with higher average µ χ . Considering the scale of our problem, where shifts from the ideal occupation are closer to 0.1, and direction plays a significant role, one would not expect a significant difference in results upon increased iterations. The three qubits with the lowest two-qubit gate error and the correct ordering of the qubit algorithm was used for the device connectivity.

Quantum Computer Calibration
Calibration data from the primary and secondary algorithms as provided by IBM's quantum computer is presented in Table 1 [35].

Electronic Structure Calculations
Molecular geometries for C 3 H 3 were taken from the Computational Chemistry Comparison and Benchmark Database [36], and for H 3 were calculated with Gaussian 09 with the coupled cluster singles and doubles method (CCSD) [37]. The basis set used Slater-type orbitals with three Gaussians (STO-3G). Electron integrals were obtained from General Atomic and Molecular Electronic Structure System (GAMESS) [38]. Maple [39] was used to perform a full configuration interaction (FCI) calculation with a QR method [40] for ground and excited states including their 1-RDMs.

Data Availability
The data that support the finding of this study are presented in Supplementary Tables 1-8. Any additional data are available from the corresponding author upon reasonable request.