Scalable Experimental Bounds for Entangled Quantum State Fidelities Shamminuj Aktar1 Andreas B artschi2 Abdel-Hameed A. Badawy1 Stephan Eidenbenz2 1Klipsch School of Electrical and Computer Engineering New Mexico State University

2025-05-03 0 0 1.02MB 19 页 10玖币
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Scalable Experimental Bounds for Entangled Quantum State Fidelities
Shamminuj Aktar1, Andreas B¨artschi2, Abdel-Hameed A. Badawy1, Stephan Eidenbenz2
1Klipsch School of Electrical and Computer Engineering, New Mexico State University
2CCS-3 Information Sciences, Los Alamos National Laboratory
Email: saktar@nmsu.edu, baertschi@lanl.gov,badawy@nmsu.edu, eidenben@lanl.gov
Abstract
Estimating the state preparation fidelity of highly entangled states on noisy intermediate-scale quantum
(NISQ) devices is important for benchmarking and application considerations. Unfortunately, exact fidelity
measurements quickly become prohibitively expensive, as they scale exponentially as O(3N) for N-qubit states,
using full state tomography with measurements in all Pauli bases combinations. However, Somma et al. estab-
lished that the complexity could be drastically reduced when looking at fidelity lower bounds for states that
exhibit symmetries, such as Dicke States and GHZ States. These bounds must still be tight enough for larger
states to provide reasonable estimations on NISQ devices.
For the first time and more than 15 years after the theoretical introduction, we report meaningful lower
bounds for the state preparation fidelity of all Dicke States up to N= 10 and all GHZ states up to N= 20
on Quantinuum H1 ion-trap systems using efficient implementations of recently proposed scalable circuits for
these states. Our achieved lower bounds match or exceed previously reported exact fidelities on superconducting
systems for much smaller states. Furthermore, we provide evidence that for large Dicke States |DN
N/2⟩, we may
resort to a GHZ-based approximate state preparation to achieve better fidelity. This work provides a path
forward to benchmarking entanglement as NISQ devices improve in size and quality.
Keywords: Fidelity, Entangled States, State Tomography, Quantinuum
1 Introduction
Any advantage that quantum computing may have over classic computing usually relies on the principle of su-
perposition, informally defined as being simultaneously in multiple computational basis states. Quantum states
that are in a superposition such that individual parts cannot be described independently of the state of others are
called entangled states. Superposition and entanglement are essential prerequisites for the successful execution of
additional gate operations per the overall structure of the quantum algorithm that leads to a desired quantum end
state. Thus, proving that a NISQ device produces entangled states is necessary for any quantum computing success.
Such low-level testing of quantum mechanical properties is necessary on quantum devices because environmental
noise quickly modifies or partially destroys a pure quantum state (as it would be theoretically prepared); this noise
process, usually called decoherence, turns the actual state in a computing device into a so-called mixed state, which
is a probabilistic mixture of our target pure quantum state and other pure states. The fidelity of a (mixed) quantum
state measures to what extent the intended target (pure) quantum state has been realized. Fidelity is defined as ‘1’
if we measure a correct pure quantum state and close to zero, i.e., 1/2Nfor a maximally mixed N-qubit quantum
state that consists purely of noise.
Let us use the state 1
√2(|00⟩+|11⟩), i.e., the Bell state as the prototypical entangled state to illustrate the concept
of fidelity. From a classical vantage point, it may appear tempting to test whether a NISQ device has produced
a Bell state as follows: We execute a large number of runs of the generating circuit, each with measurement at
the end, and count how often we measure 00, 01, 10, and 11. If we measure 00 and 11 about 50 percent of the
time each, we declare that the devices actually produce the Bell state. However, this logic is flawed because any
classical device that returns 00 and 11 with probability 0.5 would have also passed such a test. Such a cheat device
would be useless and unable to properly execute additional gate operations that propagate the quantum state and
its entanglement correctly to execute an entire quantum algorithm. An appropriate way to test entanglement is to
1
arXiv:2210.03048v3 [quant-ph] 27 Jan 2025
D2-1:(1)
D3-1:(3)
D4-1:(5)
D5-1:(7)
D6-1:(9)
D4-2:(10)
D5-2:(17)
D6-2:(24)
D6-3:(32)
Dicke States for N= 2, …, 6 and K= 1, …, N/2 (sorted by CNOT count)
0.0
0.2
0.4
0.6
0.8
1.0
Quantum Fidelity ⟨DN
K|ρ|DN
K⟩
Dicke State Fidelities on IBMQ Sydney & IBMQ Montreal
Experimental Fidelity IBM [Aktar et al. 2022]
Upper bound, same data [Aktar et al. 2022]
Lower bound, same data [this paper]
Lower bound, same data [Somma et al. 2006]
Figure 1: Experimental fidelities of Dicke states |DN
K⟩(by circuit complexity) on IBMQ devices based on full state
tomography data [3, Aktar et al. ]: (red) Exact fidelity computed from all 3Ntomography measurement settings,
(green) Upper bound computed only from the Z-basis measurement data. (blue) We compute fidelity lower bounds
from experimental X-, Y-, and Z-basis measurement setting subsets of their data, slightly improving on direct
application of existing lower bound techniques [31, Somma et al.](orange).
measure the distance between the actual state prepared on the device and the targeted Bell state. Here, we use the
quantum fidelity measure, formally defined in Section 2. Characterizing the prepared state, generally a mixed state
and thus described by a density matrix, is called quantum state tomography [4,13]. It usually requires repeatedly
preparing the state and measuring the outcome in all possible combinations of X-, Y-, and Z-axes. This is called
full-state tomography and requires 3Ntests for N-qubit states [27]. The resulting values are fitted to a density
matrix using techniques such as maximum likelihood estimation [20, 30] or Bayesian estimation [8], resulting in
further challenges to quantify errors [7,10].
Showing that entanglement exists on NISQ devices across more than ∼6 qubits in such a way is computationally
prohibitive due to the exponential number of tests. Nevertheless, it remains vital to demonstrate that the device
actually leverages quantum mechanical principles. A solution to this dilemma is to approximate or compute upper
and lower bounds on the fidelity measure, requiring fewer runs. Somma et al. [31] discovered such lower bounds
for certain symmetric entangled states based on angular momentum operators, with the additional advantage of a
smallish polynomial sampling overhead to account for errors given only by statistics of the involved measurements.
In this paper, we adopt and modify these generic bounds for Dicke states and GHZ states – three well-known
classes of entangled states – as well as approximate Dicke states. Dicke states are equal amplitude superpositions
of all computational states of the same Hamming weight (i.e., number of ones). GHZ states are Bell states that
are generalized to larger qubit counts. Both Dicke & GHZ states are studied extensively because of their quantum
mechanical properties and their use in various application domains, such as combinatorial constraint optimization.
Through a series of NISQ experiments, we show that the bound from [31] has become useful in practice to
provide evidence for entanglement across up to ‘20’ qubits without resorting to exponentially expensive full-state
tomography. This is a testament to advances in state preparation algorithms and improvements in NISQ hardware.
Figure 1 shows that the original bounds were not yet useful for NISQ devices, even as recent as 2022 for Dicke
states on as few as ‘5’ qubits (as they give negative values). They become slightly positive with our modifications
2
without providing strong evidence of entanglement; however, as we will show, newer devices (Quantinuum) produce
results that are good enough for these bounds to become meaningful lower bounds for fidelity. The original and our
modified fidelity bounds require only very few different measurement runs (three measurement settings for Dicke
states and two for GHZ states), thus making fidelity-based entanglement verification easy on NISQ devices, even
at high qubit counts. The contributions of the paper are as follows1:
•We propose an improved divide-and-conquer based method for efficient Dicke state preparation with O(N)
circuit depth and O(KN) gate count with low constant factors.
•We utilize and improve the lower bounds on the quantum fidelity from theory [31] using only three measure-
ment settings for Dicke states and only two for GHZ states.
•We demonstrate, using large-scale experiments on Dicke and GHZ state preparation on Quantinuum’s H1
ion-trap quantum device, the usefulness of the bounds for the first time on real quantum hardware.
•We report meaningful lower bound for all Dicke States up to ‘10’ qubits and all GHZ states up to ‘20’ qubits
using efficient circuit implementations.
•We give state preparation fidelities that match or exceed exact fidelity records. For example, we provide state
preparation fidelity lower bounds of (i) 0.46 for the ‘10’ qubit Dicke State |D10
5⟩and (ii) 0.73 for the ‘20’ qubit
GHZ State |G20⟩. These match or exceed exact fidelity records recently achieved on IBM’s superconducting
systems for much smaller states |D6
3⟩[3], and |G5⟩[12], respectively.
•Additionally, we show an alternative method for preparing larger Dicke state |DN
K⟩approximately, using
product states |P SN
K⟩for small Kand Even/Odd Hamming weight states |EN⟩,|ON⟩for large Kwith
even/odd parity.
•We show those approximate states can give a good approximation of larger Dicke states by computing lower
bounds for product state |P SN
K⟩up to ‘10’ qubits & Hamming weight N/2 and odd/even Hamming weight
states |ON⟩,|EN⟩up to ‘10’ qubits.
The rest of the paper is organized as follows: Section 2 summarizes the related works on fidelity estimation.
Section 3 includes our experimental methodology. Sections 4 & 5 review the state preparation, lower bound
estimation, and experimental results for Dicke and GHZ states. Section 6 includes product state and odd/even
state preparation as approximate Dicke state, lower bound computation, and their experimental results. Section 7
concludes the paper.
2 Related Work
We use the quantum fidelity Fas a similarity measure between a prepared mixed state ρ, expressed as a density
matrix, and a target pure quantum state ρψ=|ψ⟩⟨ψ|[22]:
F(ρψ, ρ) = Trq√ρρψ√ρ2
= Tr(ρψρ) = ⟨ψ|ρ|ψ⟩(1)
where √ρdenotes the unique positive semi-definite square root of ρsuch that √ρ√ρ=√ρ†√ρ=ρ, where the first
equality would also hold for a mixed state ρψ.
In this paper, we consider pure target states, notably Dicke states |DN
K⟩, and GHZ states |GN⟩. We remark that
quantum fidelity is sometimes also defined as F′=√F[27,31]. In a straightforward way, the fidelity of a prepared
N-qubit state can be computed by sampling ρin 3Ndifferent Pauli basis (N-fold tensor products of Pauli operators
σx,σyand σz) to reconstruct its density matrix; hence, we can compute the fidelity with the target state.
1Parts of our experimental bounds on Dicke and GHZ state fidelity have appeared in an extended abstract at ACM Computing
Frontiers 2023 [2].
3
Suppose we want to upper bound the quantum fidelity. In that case, we can consider the measurement success
probability,i.e. , the overall probability of sampling non-zero amplitude states when measuring in the computational
Z-basis:
MSP(ρψ, ρ) := Xx∈{0,1}n,⟨x|ρψ|x⟩̸=0 ⟨x|ρ|x⟩(2)
Another upper bound measure is Hellinger Fidelity, which quantifies the similarity between the probability distri-
butions of the pure target state and the measurement probabilities in the computational Z-basis:
H(ρψ, ρ) :=
X
x∈{0,1}Nq⟨x|ρψ|x⟩·⟨x|ρ|x⟩
2
(3)
Using a double application of the Cauchy-Schwarz inequality, one gets F(ρψ, ρ)≤H(ρψ, ρ)≤MSP(ρψ, ρ) [3].
Several works have shown that it is possible to estimate a lower bound on the quantum fidelity using only a
few measurement settings, avoiding full-state tomography. Earlier works [17, 31] focused on certain states with
unique symmetry that require only a small random subset of Pauli operators to estimate fidelity. Somma et al. [31]
proposed expressions for estimating quantum fidelity for highly symmetric classes of multi-qubit state preparation,
i.e., rotational invariant states, stabilizer states, and generalized coherent states. To estimate the lower fidelity
bound, they only used the measurements related to the symmetry operators of the density matrix. Guhne et
al. [17] derived fidelity estimation for GHZ and Wstates using 2N−1 measurements. Flammia et al. [15] had
a generalized approach where they showed estimation techniques for all possible state preparation by measuring
fewer Pauli operators close to the desired state and of greater importance. Mentioning the scalability issue in
generalized Nqubit state fidelity estimation, Elben et al. [14] showed how randomized measurement could enable
fidelity estimation by comparing two states. Zhang et al. [39] proposed a machine learning (ML) approach for direct
fidelity estimation as a classification problem using a constant number of expectations. Recently, a classical shadow
approach has been introduced to enhance tomography efficiency, with potential benefits for fidelity estimation for
certain quantum states [19, 33].
Additionally, quantum state verification (also known as the non-tomographic method) uses advanced statistical
approaches to verify whether the output of some device is the target state. Previous works on quantum state
verification techniques validate specific state preparation such as Dicke states [25], GHZ states [24], Hypergraph
states [40], Pure states [23, 35, 36, 41], etc. There are also some generalized approaches for state verification [21, 28,
34,38,42, 43]. Furthermore, Yu et al. [37] utilizes advanced statistical methods for resource-efficient quantum state
verification.
3 Methodology
In this work, we run the experiments on both the hardware and emulator devices of Quantinuum H-systems [1]
using the Quantinuum Python API. Running jobs on the Quantinuum backend requires H-System Quantum Credits
(HQC) that are linear with both the number of shots per circuit and the number of CNOTs in the circuit. The
total cost generally scales with the product of the number of shots and CNOT counts. Our initial goal was to
determine the number of experiments and the number of shots for different state preparations permissible within
the HQCs allocated to us. As the Quantinuum machines are available through a calendar month and HQCs are
allocated monthly, we target to complete experiments throughout several months with available hardware credits.
First, we run the experiments on the Quantinuum H1-2E emulator to estimate the confidence interval sizes we
can expect from the quantum hardware. To see the effect of the number of samples/circuit executions, we compute
the cumulative bounds for fidelity and measurement success probability. Similarly, we compute a two-sided 68%
confidence interval below and above the estimator for the mean based on Student’s t-distribution to check the
bounds’ stability with an increasing number of shots. Analyzing the results of emulator experiments, we get an
approximation of a meaningful number of experiments that can be performed on quantum hardware for various
states, improving on the (polynomial) number of samples derived from theoretical considerations (as elaborated on
later). We limit Dicke state preparation up to N= 10 as Dicke States starting at N= 11 become prohibitively
expensive computationally. For GHZ states, we go up to N= 20 as we run the experiment on a 20-qubit H1-1
4
摘要：

ScalableExperimentalBoundsforEntangledQuantumStateFidelitiesShamminujAktar1,AndreasB¨artschi2,Abdel-HameedA.Badawy1,StephanEidenbenz21KlipschSchoolofElectricalandComputerEngineering,NewMexicoStateUniversity2CCS-3InformationSciences,LosAlamosNationalLaboratoryEmail:saktar@nmsu.edu,baertschi@lanl.gov,...

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