NNeutrino Oscillations in a Quantum Processor
C.A. Arg ¨uelles and B.J.P. Jones Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA
ABSTRACT
Quantum computing technologies promise to revolutionize calculations in many areas of physics, chemistry, and data science.Their power is expected to be especially pronounced for problems where direct analogs of a quantum system under study canbe encoded coherently within a quantum computer. A first step toward harnessing this power is to express the building blocksof known physical systems within the language of quantum gates and circuits. In this paper, we present a quantum calculationof an archetypal quantum system: neutrino oscillations. We define gate arrangements that implement the neutral lepton mixingoperation and neutrino time evolution in two-, three-, and four-flavor systems. We then calculate oscillation probabilities bycoherently preparing quantum states within the processor, time evolving them unitarily, and performing measurements inthe flavor basis, with close analogy to the physical processes realized in neutrino oscillation experiments, finding excellentagreement with classical calculations. We provide recipes for modeling oscillation in the standard three-flavor paradigm as wellas beyond-standard-model scenarios, including systems with sterile neutrinos, non-standard interactions, Lorentz symmetryviolation, and anomalous decoherence.
The unexpected and Nobel Prize-winning discovery of neutrino oscillations has led to a program of experiment and theorythat has shaped our understanding of the role of neutrinos in the Universe. The spontaneous transition of neutrino flavor overmacroscopic distances, a phenomenon known as neutrino oscillations due to its periodic behavior, demonstrates that neutrinoshave masses that are non-zero but uniquely small. This smallness suggests connections to high-scale physics , and may berelated directly to the predominance of matter over antimatter abundances in the Universe . Studies of neutrino oscillationshave thus contributed and will continue to contribute greatly to our understanding of nature.Experiments measure neutrino oscillations by studying a neutrino beam’s flavor composition at different energies E andbaselines L . Oscillation refers to spontaneous transformation between the three neutrino flavors – ν e , ν µ , and ν τ – during flight.This is due to de-phasing of the neutrino wave functions during propagation due to a misalignment between the flavor and massbases. In the absence of strong matter interactions, and when two neutrino mass states dominate the oscillation, a sinusoidalflavor variation as a function of L / E is characteristic. For oscillations in matter and with three neutrinos participating, morecomplex functional forms are observed . Well-studied neutrino sources, in which neutrino flavor changing has been observed,include ν µ and ¯ ν µ production by decays of charged pions from accelerators or in cosmic-ray air showers , productionof ¯ ν e by fission in nuclear reactors , and of ν e by nuclear fusion in the Sun . Neutrino oscillations have been shown toviolate the Leggett-Gaarg inequality , a time domain version of Bell’s classic argument , which illustrates that they are a trulyquantum mechanical phenomenon with no possible description in terms of hidden classical variables.For neutrino oscillations to be observable, quantum coherence between the neutrino mass basis states must be maintainedover the flight distance of the neutrino , which in some experiments is thousands of kilometers. Neutrinos are thus very-long-baseline quantum interferometers, and have they been used as such to perform fundamental tests of quantum mechanics andLorentz invariance , in order to search for evidence of quantum gravity and violations of the equivalence principle .The expected decoherence of oscillating neutrinos via wave-packet separation has been studied theoretically , but not yetobserved in experiments.The apparent ease with which neutrinos avoid decoherence over thousands of kilometers of travel is undoubtedly enviableby the developers of technologies for quantum computing . Quantum processors must maintain coherence between severalentangled quantum-bits (qubits) for long durations in order to employ complex quantum algorithms without error accumulationthrough decoherence or noise. Recently, publicly accessible quantum processors were made available online as part of the IBM-Q project, and these can be used for novel research into quantum processing . Although the technology remainsimperfect, with error rates per gate operation of O ( . ) and per qubit read of O ( ) prohibiting very lengthy calculations,the platform provides a test bed for exploring quantum solutions to computational problems, and finding ways to re-expresscalculations in the language of quantum circuits. Quantum processors are expected to provide the most immediate benefitsfor analyzing systems that are fundamentally quantum mechanical in nature . Examples of physics frontiers that may besubstantially advanced by quantum computation include modeling in nuclear physics , collective oscillations of neutrinos insupernova explosions , many-body effects in condensed matter systems , and quantum chromodynamics , among others.1 a r X i v : . [ qu a n t - ph ] M a y n this paper, we present a quantum simulation of neutrino flavor oscillations. After illustrating how to encode the two-neutrino system evolution in a quantum computer with a single qubit, we proceed to implement the less intuitive three-neutrinosystem realized on a subspace of a two-qubit Hilbert space. The primary challenges involved are the implementation of thePontecorvo–Maki–Nakagawa–Sakata (PMNS) operation, which relates the flavor and mass neutrino eigenstates, and thetime-evolution operator of the system in the computational basis. After testing that our quantum circuit reproduces the quantumneutrino oscillation probability on the IBM-Q Yorktown public quantum computer, we conclude with a brief discussionof how to include more complex phenomena including sterile neutrinos, matter effects, non-standard interactions, Lorentzsymmetry violation, and decoherence within the quantum algorithm.
Two-flavor neutrino oscillation
Two-flavor neutrino oscillations involve a Hilbert space of two dimensions. This can be represented on a single qubit, via thebasis choice: | (cid:105) = | ν (cid:105) = (cid:18) (cid:19) and | (cid:105) = | ν (cid:105) = (cid:18) (cid:19) . (1)The rotation into the flavor basis requires a unitary operation via the two-dimensional PMNS matrix. The reduced PMNSoperation is defined such that | ν e (cid:105) = U x PMNS | (cid:105) and | ν µ (cid:105) = U x PMNS | (cid:105) . The most general unitary transformation applicable to asingle qubit system, which must be able to support the 2 × U U ( Θ , φ , λ ) = (cid:18) cos Θ − sin Θ e i λ sin Θ e i φ cos Θ e i ( λ + φ ) (cid:19) . (2)For the two-neutrino system, oscillation probabilities depend only on one of the parameters of U
3, for the following reasons :1. The parameter φ can be removed by a re-defininition of the | ν µ (cid:105) basis state via | ν µ (cid:105) → e − i φ | ν µ (cid:105) . This corresponds tore-phasing the charged muon field, under which the Standard Model Lagrangian is invariant. Without loss of generality,we can set φ = λ could similarly be removed by re-phasing the | ν (cid:105) field, | ν (cid:105) → e i λ | ν (cid:105) . The Lagrangian is onlyinvariant under this re-definition if the neutrinos are Dirac particles. If they are Majorana particles, on the other hand, thisphase is physical and must be maintained in the Lagrangian. However, it can be shown the “Majorana phase” λ does notinfluence neutrino oscillations and general oscillation probabilities can be calculated under the assumption λ = × † , via the definition: U x = U ( θ , , ) = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) and U x = U ( − θ , , ) . (3)To prepare a neutrino flavor state, we can apply the PMNS operation either to the | (cid:105) state to prepare a | ν e (cid:105) , or to the | (cid:105) state to prepare a | ν µ (cid:105) . The input qubits in a quantum computation conventionally initialize to | (cid:105) , and the | (cid:105) state can beprepared by application of the Pauli-X gate, | (cid:105) = X | (cid:105) . The preparation of electron and muon neutrino flavor states, as well as m and m mass states in the two-flavor basis is shown in terms of quantum circuit elements in Fig. 1.Oscillation probabilities can be calculated by time-evolving the initial flavor state vector in with the appropriate time-evolution operator U and then measuring in the flavor basis. Only relative phases between mass states are relevant foroscillations, and so without loss of generality, we can measure all phases relative to the m basis state. The time-evolutionoperation can thus be encoded in an S gate: U ( t ) = S ( φ ) = (cid:18) e i φ (cid:19) , (4)where φ = ∆ m t / E ¯ h . In the two-flavor system, we thus find a particularly simple representation of the PMNS and time-evolution “gates”, shown in Fig. 1 center and right. Examples of circuits that realize various two-flavor oscillation scenarios aregiven in Fig. 2, left. With the quantum circuit defined, we can proceed to evaluate oscillation probabilities on the quantumprocessor. We run 1024 trials and count flavor measurement outcomes to establish oscillation probabilities in the two-flavorsystem. Fig. 2, right, shows the comparison of the quantum calculation to the theoretical expectation for parameters relevant to ν e disappearance at the KamLAND experiment as an example. The figure compares actual quantum computations, calculatedon the IBM quantum processor (squares), simulated runs of the quantum computer, which represent the same operationsperformed without decoherence or errors (dots), and the standard two-flavor oscillation formula (lines). The quantum evolutionmatches very well with expectations from both theory and quantum simulation. ν ν e ν μ PMNS 𝒰 (t) Two-neutrino PMNS Gate tΘ Two-neutrino time evolution gateTwo-neutrino state preparation
PMNS † PMNS † XX U3(2θ) U1(t)
PMNS 𝒰 (t) Three-neutrino PMNS Gate Three-neutrino time evolution gate
U1(at)U1(bt)U3( 𝜶 ) U3( 𝛄 ) U3( 𝜺 )U3(β) U3( 𝜹 ) U3( 𝛇 ) Figure 1.
Top: Preparation of neutrino flavor and mass basis states in the 2 × × Three-flavor neutrino oscillation
A three-flavor neutrino oscillation involves a Hilbert space of dimension three, requiring more than one qubit. The minimalrepresentation can be encoded on two qubits, via a basis definition such as: | (cid:105) → | ν (cid:105) = , | (cid:105) → | ν (cid:105) = , | (cid:105) → | ν (cid:105) = , and | (cid:105) → | ν X (cid:105) = . (5)There is one redundant basis state, | ν X (cid:105) , in this representation. This could represent a fourth neutrino flavor in models withsterile neutrinos, but for the present example we will consider it as physically decoupled, and thus unphysical. As in thetwo-flavor case, to prepare a flavor state, we must apply the PMNS operation to an initial state in the computational basis: | ν e (cid:105) = U x | ν (cid:105) , | ν µ (cid:105) = U x | ν (cid:105) , and | ν τ (cid:105) = U x | ν (cid:105) . (6)Unlike in the two-neutrino example, however, creating a set of quantum gates to implement the PMNS operation on twoentangled qubits is non-trivial. A real unitary two-qubit gate requires at least two CNOT and 12 elementary gates for anentirely general representation. Constraints on the PMNS matrix due to re-phasing invariance may be expected to allow fora more compact representation. Following exploration of several possibilities, we constructed a parameterizable set of sixreal U3 gates acting on two qubits A and B , with two interspersed CNOT gates to reproduce the PMNS operation. To fix thefree parameters of this arrangement, we map the circuit onto matrix multiplication in the computational basis, and performa numerical fit to match its entries to the experimentally determined PMNS matrix elements . We independently fit for thePMNS matrix and for its adjoint, which are both needed for oscillation computations. The PMNS and PMNS † operations arethus constructed as:PMNS = U A ( ε ) U B ( ζ ) CNOT AB U A ( γ ) U ( δ ) B CNOT AB U A ( α ) U ( β ) B , (7)PMNS † = U B ( ε (cid:48) ) U A ( ζ (cid:48) ) CNOT AB U A ( γ (cid:48) ) U B ( δ (cid:48) ) CNOT AB U A ( α (cid:48) ) U ( β (cid:48) ) B . (8)The best-fit parameters α , β , γ , δ , ε , and ζ (primed and unprimed) reproduce the measured PMNS and PMNS † elements withinone part in 10 when no CP violating phase is present– comfortably within experimental uncertainty. This parameterizationcan be extended with two additional U3 gates and a CNOT in order to incorporate a Dirac CP phase with similar accuracy. e disappearance ν μ à ν e ν μ à ν 𝛕 Prepare flavor state Rotate to mass basis Time evolution Rotate from mass basis Measure
Figure 2.
Left: Quantum circuit embodying two-flavor neutrino oscillation. Right: Two-flavor electron-neutrino survivalprobability as a function of the neutrino energy. The green line shows the theoretical calculation using a classical computer.The black circle markers indicate a quantum computer simulation and the black square markers are the output of the IBM Qquantum computer.These parameters are tabulated in the Methods section. The PMNS gate decomposition in terms of component gates is showndiagrammatically in the bottom left panel of Fig. 1.Once the initial flavor state is prepared, the time-evolution operation U must be implemented. This is represented in thecomputational basis by U ( t ) = exp (cid:2) i diag (cid:0) , ∆ m t E ¯ h , ∆ m t E ¯ h , Φ (cid:1)(cid:3) , where Φ is an arbitrary phase that can be picked forconvenience, since the fourth basis state is unobservable. A straightforward choice that can be implemented as one-qubit gatesacting on A and B is: U ( t ) = S A (cid:16) i ∆ m t E ¯ h (cid:17) S B (cid:16) i ∆ m t E ¯ h (cid:17) ; (9)where S A and S B are the S gates for qubits A and B , respectively.The complete quantum circuit for the oscillation calculation in the three-neutrino space comprises of state preparation,time-evolution and flavor measurement, and is shown in Methods Fig. 5. This circuit can be run repeatedly to prepare qubits inflavor eigenstates, time-evolve them, and measure their flavor after propagation in order to establish oscillation probabilities.Since this is a substantially more complex circuit than the two-flavor case, gate errors and read errors are expected to be moreprevalent. We correct for the effects of read errors in the final oscillation probability by applying an inverted error matrix in thecomputational basis, which accounts for decoherence and read errors in a statistical manner, based on qubit readout accuracymeasured using runs with L / E =
0. More details on this procedure can be found in the Methods section.Figure 3 shows two example calculations of three-flavor oscillation probabilities given an initial muon neutrino beam. Thefirst panel shows calculations at smaller L / E where the oscillation is effectively a two-flavor system. The second panel showsthe behavior near the first oscillation maximum where three flavors ν e , ν µ , ν τ , are participating strongly. Good agreement withtheory is observed in both regimes. In both cases, the electron flavor is slightly over-represented, potentially due to read andgate errors that are not entirely symmetrically distributed between flavors. The size of the effect is comparable to the statisticaland systematic uncertainty, which receives contributions from: 1) accumulated gate errors, based on the ibmqx2 spec of ∼ − per gate added in quadrature over 50 gates and 2) statistical uncertainty, from the finite number of evolutions (1024)used to establish the oscillation probability. After running the simulation and applying statistical error correction tuned in thesideband region of the calculation, strong agreement between the quantum and classical computations are obtained. igure 3. Calculations of three-flavor neutrino oscillations evaluated using a quantum computer (squares), quantum computersimulator (circles), and theory (lines). The quantum computer results have been corrected for gate read errors based on thematrix M described in the text. The two plots show three-flavor oscillation probabilities calculated in two characteristic L / E ranges. Neutrino oscillations with new physics
In addition to standard neutrino oscillations, beyond-standard-model effects that have been searched for in neutrino oscillationexperiments can be incorporated into the quantum circuit straightforwardly, by either a) extending the PMNS matrix andtime-evolution operator to higher dimensionality or b) introducing new effects in the time-evolution term. Here we brieflyreview a few of these scenarios.
Sterile neutrinos
As we have seen, the incorporation of at least one additional basis state within the Hilbert space is mandatory, given a two-qubitrealization. To use this state to represent an oscillating forth neutrino, as suggested by short baseline neutrino anomalies ,two adjustments are required: 1) extension of the PMNS matrix to mixing in four dimensions, which is already achievable inour present parametrization, given appropriate gate coefficients; and 2) independent control of the phase of the | ν (cid:105) = | (cid:105) mass state in the time evolution operator. This is necessarily an operation that involves entangling the two qubits, and so cannotbe implemented on single-qubit gates only. A circuit that produces the required time evolution (an independently specifiedphase on each of | (cid:105) , | (cid:105) , and | (cid:105) ) is shown in Fig. 4, top. This circuit configured with parameters φ = E ¯ h (cid:0) ∆ m − ∆ m + ∆ m (cid:1) , φ = E ¯ h (cid:0) ∆ m + ∆ m − ∆ m (cid:1) , and φ = E ¯ h (cid:0) − ∆ m + ∆ m + ∆ m (cid:1) (10)will achieve the necessary four-state time evolution needed to implement quantum simulations of the four-flavor neutrinosystem extended for a single sterile neutrino. Non-standard interactions and matter effects
The modelling of either standard or non-standard matter effects can be incorporated without changing the three-flavoroscillation quantum circuit, by adjustment of the input parameters that describe the PMNS and time-evolution operations in themodified matter basis. A discussion of the parametrizations that incorporate these effects is given in the Methods section.
Decoherence
Decoherence is a non-standard neutrino oscillation effect that is often considered in connection with quantum gravityor spacetime foam models . In decoherence scenarios, development of entanglements between parts of the neutrino wave )U1(ϕ ) U1(ϕ ) 𝒰 (t) Sterile neutrino time evolution
U1(at)U1(bt) U3(dt) U3(-dt) 𝒰 (t) Decoherent oscillations time evolution
Figure 4.
Quantum circuits for two beyond-standard-model oscillaion scenarios. Top: sterile neutrino oscillations; Bottom:anomalous decoherence in the mass basis.function and an external environment lead to partial collapse of the wave function and suppression of oscillations. Decoherencecan be manifest in various bases, depending on the degrees of freedom within the neutrino subsystem that the environmententangles with. Fig. 4, bottom, illustrates a quantum circuit that implements decoherence in neutrino oscillations via generationof entanglements in the mass basis for small dt . In this circuit, an auxiliary qubit representing the environment acquires a smalland time-dependent admixture of the | (cid:105) basis state, if and only if the second neutrino qubit is in a | (cid:105) state. Measurement ofthe environment qubit creates partial decoherence in the neutrino system. This example could model gravitational decoherence,or of flavour change through wave packet separation given a normal ordering of neutrino masses where one mass state is muchheavier than the others. For large dt , the second qubit acquires a sufficient phase shift to make a full rotation and restorecoherence. The process of continuous measurement during evolution for large dt is implemented by repeated circuit units oftime evolution, entanglement and measurement operations. Conclusions
We have demonstrated a quantum mechanical simulation of neutrino oscillations on a quantum computer, using both two-flavorand three-flavor systems. The two-flavor system has an almost trivial realization in the two-dimensional Hilbert space of asingle qubit, with implementation of PMNS and time-evolution gates using individual single-qubit gates. The three-neutrinosystem, on the other hand, requires a more complicated quantum circuit, involving the entanglement of two qubits to produce aHilbert space of four dimensions. A subspace of three of these states is used for the calculation. Our implementation of the3 × ν and allowing for phase freedom in the unphysical fourth basis state, the time-evolution operator can beimplemented using two single-qubit gates.Quantum calculations using both the two- and three-flavor system agree with theoretical expectations for the neutrinooscillation probability within systematic and statistical uncertainty. Although in agreement, the calculation presented hereis characteristically different to the classical computation of oscillation probabilities, since the qubits act as direct quantumanalogues to the evolving neutrino flavor wave function. The system is prepared coherently, evolves forward in time unitarily,and has its wave function collapsed to measure the final flavor oscillations, just as in a neutrino oscillation experiment.Analogues of real quantum systems inside quantum processors such as the one presented in this paper may eventuallyenable computations that surpass the capabilities of their classical counterparts. This is especially likely for strongly coupledor highly entangled systems, such as collectively oscillating neutrinos in supernovae, for example. Understanding how totranslate simple and well-understood calculations into quantum circuits is a necessary first step toward realizing this goal. Inthis work, we have presented one such example, creating an analogue to two and three-flavor neutrino oscillations inside apublicly accessible quantum processor. This may be the first of many neutrino oscillation calculations to profit from the powerof quantum information processing technologies. ethods Running quantum computation on ibmqx4
Computations were run on the IBM publicly accessible quantum computer ibmqx4 (Yorkville). The two least read-error-pronequbits on this five qubit machine that could be connected by the appropriate logic gates within the allowable topology werechosen to support the computational basis. At the time of writing these were qubits 0 and 2, with reported gate errors of0.77 × − and 1.03 × − , and read errors of 7 .
6% and 2 .
9% respectively, and a mulit-qubit read error of 2.21%.It is difficult to convert such gate-wise error specifications into an expected calculational accuracy, so we instead opted tomeasure the accuracy directly by running L / E = ibmqx4 is shown inSupplementary Fig. 5. This was run with 1024 shots to establish the survival probability, and the statistical error associatedwith this count is included in our uncertainty budget.As well as access to the quantum processor, the IBM website offers a quantum simulator tool to simulate the circuit beforerunning it. In all cases, simulations agreed near perfectly with theoretical expectations. This suggests that wherever the datashow small differences from theoretical expectations, these are to be attributed to the imperfections of the quantum processor. Fitting the PMNS matrix gate parameters
The fit to the PMNS matrix is made by gradient minimization over six parameters. We minimize the sum of squared residualsof each of the elements of the matrix. The final fit converges to the true PMNS matrix within one part in 10 for every element,comfortably within experimental uncertainty. The input values of the PMNS matrix used in the fit are from and are: U PMNS = . . . − . . . . − . . (11)The bottom row represents a decoupled, unphysical state, but is necessary to span the four-dimensional Hilbert space representedby two qubits. The best-fit parameters from fitting the PMNS and PMNS † matrices in the computational basis are: α = − . β = . γ = . δ = . ε = . ζ = − .
065 (12) α (cid:48) = − . β (cid:48) = − . γ (cid:48) = . δ (cid:48) = − . ε (cid:48) = . ζ (cid:48) = . Correcting for bit flips and decoherence
Some random qubit readout errors are naturally expected for any quantum computation. However, we can correct our finaldistributions against some of these information losses statistically. We consider that there is some average probability f and f MeasurementPMNSPMNS † Construct ν μ 𝒰 (t) Figure 5.
Three-flavour neutrino oscillation experiment as run on the IBM quantum computer. or either of the qubits to be flipped leading to an incorrect flavor measurement, which is approximately uniform across circuits.Then, the effect on the final distribution of events in ( ν e , ν µ , ν τ , ν X ) space is to multiply final state distributions by a matrix M: M = ( − f ) ( − f ) + f + f ( − f ) f − f ( − f ) f − f f f ( − f ) f − f ( − f ) ( − f ) + f + f f f ( − f ) f − f ( − f ) f − f f f ( − f ) ( − f ) + f + f ( − f ) f − f f f ( − f ) f − f ( − f ) f − f ( − f ) ( − f ) + f + f (14)The form of M can be understood by considering that spurious transitions ν µ ↔ ν e , ν τ ↔ ν e , ν τ ↔ ν s and ν µ ↔ ν s requireonly one qubit flip in the final read, whereas spurious transitions ν e ↔ ν s , ν µ ↔ ν τ require two. f and f , the rates of bit flipsin qubits 1 and 2, can be measured by examining the rate of spurious transitions at L / E = L / E =
0, whereas we find the quantum computer gives random some transformations, consistent with f ∼
13% and f ∼ f measured, we can invert M and apply this inverted matrix to correct the final probabilitydistributions. This correction is applied in order to obtain our final data / theory comparisons. Incorporation of matter potentials and non-standard interactions
When neutrinos travel through a medium they experience a potential produced by coherent-forward scattering with electrons,protons, and neutrons. The potential sourced by protons and neutrons is the same for all neutrino flavors, and thus inducesan overall phase in the neutrino system evolution. The electron charged-current potential is flavor asymmetric, producing anobservable modification in the neutrino oscillation probability. The matter potential can be written in the flavor basis as V m = √ G F N e , (15)where G F is the Fermi constant and N e is the electron number density. Then the total neutrino Hamiltonian can be written as H = H vac + V m = U † m ∆ U m . (16)The Hamiltonian can be diagonalized by a unitary transformation, U m , that relates the flavor basis to the Hamiltonian eigenstates,and a diagonal matrix, ∆ , which contains the energy eigenvalues. In a two flavor system, U m can be written as a 2 × θ m = arctan ∆ sin θ ∆ cos 2 θ − √ G F N e , (17)where ∆ = ∆ m / ( E ) and θ is the vacuum mixing angle. The relevant Hamiltonian eigenvalue difference is given by λ = (cid:114)(cid:16) ∆ cos 2 θ − √ G F N e (cid:17) + ∆ sin θ . (18)Thus the matter modification does not require a new quantum circuit; we can simply replace the vacuum mixing angle andeigenvalue for the expressions given above. For the case of the three-neutrino scenario exact expressions for the effectivemixing angles and eigenvalues are lengthy, but can be readily found by numerical diagonalization.Effects of the standard neutrino matter potential have been observed with natural sources, e.g. in Solar neutrino experiments,and with human-made sources, e.g. in accelerator neutrinos experiments. Deviations from the standard potential can be due tonew forces that manifest themselves as vector or scalar interactions. These have been constrained by searches of anomalousneutrino flavor changing and, more recently, in coherent-scattering neutrino experiments. The status of recent constraint onnon-standard interactions can be found in Ref. , where the constraints are given relative to the weak-force strength, G F , andflavor dependent coefficients, ε αβ . Depending on the target and the flavor structure these are constrained from O(1%) toO(10%). The effects of vector non-standard interactions can also be calculated using a quantum processor by using PMNS gatewith effective matter potential mixing angles. The appropriate angles and eigenvalues can be determined by diagonalizing thefollowing Hamiltonian in (16) where one ought to replace V m by V nsim = √ G F N e + ε ee . (19) ncorporation of Lorentz Violation The Standard Model (SM) of particle physics can be thought as an effective field theory towards a grand unified theory ofnature . In many extensions of the SM, Lorentz symmetry is broken. Neutrinos, as natural interferometers, are extremelysensitive to high-scales where Lorentz Violation (LV) may be manifest. In fact, neutrinos have some of the strongest constraintson LV non-renormalizable operators . Calculations of oscillation probabilities in the presence of LV can be performed usingthe quantum circuits presented in this paper. LV can be incorporated in the neutrino Hamiltonian in the following way: H = H vac + ˜ a + ˜ c E + ˜ a E + ˜ c E + ..., (20)where ˜ a d ( ˜ c d ) is a matrix that contains the strength of interaction between the neutrino and a Lorentz violating field producedby a CPT even (odd) effective operator of dimension d . Similarly to the case of matter interactions, this Hamiltonian can bediagonalized numerically in order to obtain appropriate effective mixing angles and frequencies. One can then use the PMNSquantum gate and evolution operators discuss in the main text. Acknowledgements
We thank Jean DeMerit, Kareem R. H. A. M. Farrag, Roxanne Guenette, and Jonathan Asaadi for proof-reading this manuscriptand Peter Denton for comments on the pre-print.BJPJ is supported by the Department of Energy under award numbers DE-SC0019054 and DE-SC0019223. CAA issupported by U.S. National Science Foundation (NSF) grant No. PHY-1801996. BJPJ and CAA thank the organizers of theworkshop “New Opportunities at the Next Generation Neutrino Experiments” at the University of Texas at Arlington, where thiswork was completed. We thank IBM for making the public quantum processor available to the world- an invaluable resource aswe prepare to move towards the quantum revolution. We acknowledge use of the IBM Q for this work. The views expressed arethose of the authors and do not reflect the official policy or position of IBM or the IBM Q team.
Author contributions statement
BJPJ and CAA made equal contributions to all aspects of this work.
Additional information
The authors declare no competing interests.
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