Optimal depth and a novel approach to variational quantum process tomography
Vladlen Galetsky Pol Julià Farré Soham Ghosh Christian Deppe Roberto Ferrara

IEEEexample:BSTcontrol





Optimal depth and a novel approach to variational quantum process tomography
    
Vladlen Galetsky^*, Pol Julià Farré2, Soham Ghosh^*, Christian Deppe2 and Roberto Ferrara^* 
^*Technical University of Munich

2Technical University of Braunschweig 

Email: vladlen.galetsky@tum.de, pol.julia-farre@tu-braunschweig.de, soham.ghosh@tum.de,  
 christian.deppe@tu-braunschweig.de, roberto.ferrara@tum.de

    Received XX XX 2024; accepted XX XX XX
==================================================================================================================================================================================================================================================================================================================================


In this work, we present two new methods for Variational Quantum Circuit (VQC) Process Tomography onto n qubits systems: PT_VQC and U-VQSVD. 

Compared to the state of the art, PT_VQC halves in each run the required amount of qubits for process tomography and decreases the required state initializations from 4^n to just 2^n, all while ensuring high-fidelity reconstruction of the targeted unitary channel U. It is worth noting that, for a fixed reconstruction accuracy, PT_VQC achieves faster convergence per iteration step compared to Quantum Deep Neural Network (QDNN) and tensor network schemes.

The novel U-VQSVD algorithm utilizes variational singular value decomposition to extract eigenvectors (up to a global phase) and their associated eigenvalues from an unknown unitary representing a general channel. We assess the performance of U-VQSVD by executing an attack on a non-unitary channel Quantum Physical Unclonable Function (QPUF). U-VQSVD outperforms an uninformed impersonation attack (using randomly generated input states) by a factor of 2 to 5, depending on the qubit dimension.

For the two presented methods, we propose a new approach to calculate the complexity of the displayed VQC, based on what we denote as optimal depth. 




VQC, Process tomography, Optimal depth, Quantum Computation, Singular Value Decomposition.
       

Optimal depth and a novel approach to variational quantum process tomography
    
Vladlen Galetsky^*, Pol Julià Farré2, Soham Ghosh^*, Christian Deppe2 and Roberto Ferrara^* 
^*Technical University of Munich

2Technical University of Braunschweig 

Email: vladlen.galetsky@tum.de, pol.julia-farre@tu-braunschweig.de, soham.ghosh@tum.de,  
 christian.deppe@tu-braunschweig.de, roberto.ferrara@tum.de

    Received XX XX 2024; accepted XX XX XX
==================================================================================================================================================================================================================================================================================================================================






§ INTRODUCTION



 §.§ Process tomography

Quantum tomography plays an essential role in the security assessment and characterization of channel and state evolution in Noisy Intermediate Scale Quantum (NISQ) devices <cit.>. A class of promising candidates, ranging from Quantum Deep Neural Networks (QDNN) <cit.>, tensor networks <cit.>, Variational Quantum Circuit (VQC) <cit.>
and quantum compilation algorithms <cit.>, try to perform state and process tomography beyond the current classical computation capabilities.


In process tomography, certain approaches presume particular symmetries within the target circuit. For instance, tensor networks <cit.> rely on the notion of low entanglement structures. In unsupervised learning and QDNN, algorithms extract multidimensional probability distributions from raw data. The considerable expressivity of such models enables the retention of assumptions regarding high entanglement degrees in the unitary under study <cit.>.

As the system to be queried scales, state and unitary reconstruction become resource-intensive tasks, often resulting in an exponential overhead <cit.>. The poor scalability, the lengthy circuit depths and the extensive number of initializations and iterations required to minimize the objective function render VQC schemes unreliable for NISQ device applications. To mitigate these challenges, we propose in Section. <ref> a new VQC process tomography scheme called PT_VQC.





 §.§ Process tomography of unitary evolutions


Studying non-unitary channels presents a more intricate task. One proposed solution introduces convex optimization <cit.> to address this challenge. However, this approach relies on understanding the constraints of the process matrix and assumes the structure of the channel under examination. Another proposed method utilizes classical shadows <cit.>, which are based on shadow tomography techniques. However, the classical shadow post-processing becomes highly inefficient as the amount of qubits n scales up. Additionally, for Clifford shadows, prior knowledge of the subsystem is required to perform the non-unitary reconstruction.

We instead introduce a novel solution in Section <ref>: the U-VQSVD scheme. We showcase its effectiveness by executing an impersonation attack against a non-unitary Quantum Physical Unclonable Function (QPUF) channel, reconstructing the singular values and vectors of the unknown unitary evolution defining it.



 §.§ Quantum Physical Unclonable Function (QPUF)

QPUFs are hardware-based unclonable devices with an inherit randomness <cit.> that can be produced, for example, by having an imperfection within an optical crystal. Based on a set of challenges, such randomness results into a production of a unique and unpredictable set of responses and this randomized mapping can be harnessed within the realm of secure authentication.


Recent such protocols  relying on a QPUF were based on a server issuing the quantum states to a user, which could  be then authenticated afterwards using the QPUF. The quantum no-cloning theorem ensures that the issued  states of the QPUF cannot be cloned and  performing quantum process tomography on a unitary is also prohibited by requiring an exponential number of queries.

    
     Recent research <cit.> introduced a mathematical framework that, for the first time, provided a QPUF scheme with provable security against quantum adversaries, referred to as "Quantum selective unforgeability". However, according to <cit.> and <cit.>, the security notion they define under the label "Quantum existential unforgeability" is still nonexistent in a unitary QPUF. Subsequently, <cit.> established an analogous notion of existential unforgeability for their designed non-unitary channel-based QPUF.
     
     We numerically show that U-VQSVD does not break the security notions of <cit.>, while still performing better than an attacker employing random states during authentication.
     
    


§ OUR CONTRIBUTIONS


We divide our set of contributions into two main parts. The former arises from a new process tomography algorithm called PT_VQC, found in Section. <ref>. In contrast, the approach proposed in <cit.> relies on an inefficient SWAP-test algorithm to define their cost function, necessitating 4^n state initializations for a complete reconstruction of the target unitary. Such high requirements for VQC process tomography schemes render them slow and unreliable for NISQ device applications, particularly those reliant on qubit dimensionality.

We instead deliver an enhanced scheme by introducing an alternative quantum circuit architecture and redefining the cost function. Notably, we reduce the required state initializations from 4^n to just 2^n and halve for each run the required number of qubits. As a consequence, we improve the processing time of the algorithm while, in parallel, enhancing the accuracy of its gradient computation by applying the 4-term shift rule. We conducted a comparative analysis of the efficiency and cost of the PT_VQC algorithm against variational Matrix Product Operators (MPO) based on tensor networks <cit.>, as well as against QDNN <cit.>.

For the second main contribution, we introduce a new algorithm called U-VQSVD, found in Section. <ref>.
U-VQSVD is inspired by the variational quantum singular value decomposition scheme proposed in <cit.>.
 Our novel proposal allows us to learn unknown general channels by efficiently reconstructing their eigenvalues and eigenvectors up to a phase. Furthermore, we evaluate the effectiveness of our new process tomography scheme by constructing an impersonation attack on the authentication protocol of a Phase Estimation-based QPUF (PE-QPUF)  <cit.>. This assessment involves comparing the capabilities of an attacker employing random states to those of an attacker utilizing U-VQSVD to generate input states, alongside a trusted party.
 
 Finally, we discuss how to select the depth of the VQC introducing the concept of optimal depth and providing the reader with a guideline for its estimation.





 



§ METHODS




 §.§ Notation

We denote vectors as, e.g., θ̂. θ̂_̂ĵ represents the j_th block of θ̂, while θ_j denotes the j_th parameter of θ̂. Quantum gates are defined with upper case letters, e.g., RY(θ). U is the targeted  unitary and UVQC is the unitary representing the variational quantum circuit.
A chain of quantum gates is denoted via a hyphen connecting two gates, e.g., RY(θ_0)-RX(θ_1). For pure quantum states, we use the Dirac bra-ket notation |ψ⟩ and lowercase Greek letters, e.g.,  ρ and σ, for general density-matrix states. The variables t and a represent the number of target and ancilla qubits of the QPUF, respectively. We denote the chosen depth of the VQC as d.



 §.§ Optimal depth


The works of <cit.> and <cit.> relate the complexity of the VQC to the amount of qubits n and depth d via 𝒪(n^3d^2) and by creating a lower bound on the number of resources R(n)≥ poly(n), respectively. However, such claims only provide a lower bound on the circuit complexity without ensuring a high performance. To offer a more practical understanding of the VQC complexity, we no longer treat d as being independent from n, but rather expect to find some dependence d(n) for an efficient VQC performance. Recognizing this circuit behavior is essential for accurately assessing its implementability and scalability in near-term NISQ devices <cit.>. To formally introduce the optimal depth, we begin by defining the concept of variational unitary space:


Variational Unitary Space: Space formed by all possible variational unitaries U_VQC attainable by a fixed architecture of the VQC W(θ̂_̂ĵ) with p real parameters


    Γ = ⋃_θ̂∈ IR^× p{ U_VQC(θ̂)}⊆𝐔(𝐧),



 where 𝐔(𝐧) represents the unitary group of n × n matrices. 
  
  UVQC can be described by block repetitions of length d(n) which defines the depth of the VQC,

    UVQC=∏_j=1^d(n)W(θ̂_̂ĵ),

 where W(θ̂_̂ĵ) represents a combination of parametrized single-qubit and two-qubit gates. The definition of optimal depth is subsequently formulated:


Optimal depth (D_opt): Assuming ∃  d s.t. U⊂Γ, for a fixed architecture of the VQC W(θ̂_̂ĵ), D_opt is the minimal amount of layers that suffices for a fiducial reconstruction of the unitary U. More concretely, with a fixed W(θ̂_̂ĵ),


    D_opt(n)=min{d(n)}s.t. U_VQC(θ̂)≈ U.





 

For a given fixed value of 
n, each depth 
d in the VQC corresponds to the number of iterations required to achieve the target accuracy of the cost function in the PT_VQC algorithm. We search for the minimal mean number of iterations needed to attain this accuracy across 10 different target Haar random unitaries. The depth d associated to it is considered as the optimal depth D_opt. In the U-VQSVD scheme, instead of considering the mean number of iterations, the maximum number of iterations times d is recorded. This behavior is depicted in Fig. <ref> within the context of U-VQSVD (Section <ref>) for the task of reconstructing the eigenvalues and eigenvectors of 10 Haar-random unitaries.





 §.§ PT_VQC algorithm



A new method is proposed for VQC process tomography (PT_VQC). We consider for our VQC the following parameterized architecture composed of single gate blocks V(θ̂_̂ĵ) and two-qubit gate blocks T_j,


    W(θ̂_̂ĵ)= V(θ̂_̂ĵ)T_j.


UVQC(θ̂) is constructed using d(n) blocks of single-qubit gate chains, comprising RZ-RY-RZ gates, and two-qubit gate blocks T_j=CX_odd_j|even_j. Here, CX_odd_j|even_j represents a sequence of CX gates controlling either odd or even qubits, depending on the block number j.
 
 The architecture for the VQC representing the UVQC for the PT_VQC and U-VQSVD algorithms can be seen in Fig. <ref> and the design of the algorithm is presented in Fig. <ref>. For a set of initializations completing an orthonormal basis {|i⟩}_i=0^2^n-1, the state |ξ⟩ before measurement has the form



    |ξ⟩=1/2(|Ψ_i^pr⟩+|Ψ_i^tr⟩) ⊗|0⟩+1/2(|Ψ_i^pr⟩-|Ψ_i^tr⟩) ⊗|1⟩,


where |Ψ_i^pr⟩ = UVQC(θ̂)|i⟩
and |Ψ_i^tr⟩ = U|i⟩.

To define the subsequent partial measurement, the Von Neumann formalism is used, with the measurement projector  Ω_0=I⊗|0⟩⟨0| being related to the probability p_0 of obtaining the outcome "0" in the following manner


    p_0=⟨ξ|(Ω_0)^†(Ω_0)|ξ⟩
    
    =1/2 - 1/2(⟨Ψ_i^tr||Ψ_i^pr⟩). 6



















We define the cost function as 

    C(θ̂)    = 1/2^n∑_i=1^2^n|Ψ_i^pr⟩-|Ψ_i^tr⟩^27
       =
    1/2^n∑_i=1^2^n(⟨Ψ_i^pr|Ψ_i^pr⟩+
    ⟨Ψ_i^tr|Ψ_i^tr⟩
        + ⟨Ψ_i^pr|Ψ_i^tr⟩⟨Ψ_i^tr|Ψ_i^pr⟩)8
    9   = 1/2^n-1∑_i=1^2^n[1-(⟨Ψ_i^tr|Ψ_i^pr⟩)].


In Appendix. <ref>, we provide the mathematical justification for the choice of this cost function. Its computation comes from the measurement over a controlled version of the VQC, thus its gradient is computed by the 4-term shift rule <cit.>

    
    ∇_θ_iC(θ̂) =  
          √(2)+1/4√(2)[C(θ̂)|_θ_i = θ_i + π/2-C(θ̂)|_θ_i =θ_i-π/2]-
    √(2)-1/4√(2)[C(θ̂)|_θ_i= θ_i + 3π/2-C(θ̂)|_θ_i=θ_i - 3π/2].
          10


 The target unitary U is sampled from the Haar measure using the QR decomposition. In the minimization process, we employ the Adam optimizer, with hyperparameters β_1 = 0.8 and β_2 = 0.999, which respectively denote the decay rates of the first and second-moment estimates. In Appendix. <ref>, we present the pseudo-code summarizing the algorithm routines for PT_VQC.









 §.§ PT_VQC comparison to QDNN and Tensor Networks

In this section, we outline the methodology utilized for comparing PT_VQC, QDNN <cit.> and tensor networks <cit.> using MPOs. We set the number of iterations to 60 and establish the target average fidelity as


    F=1/2^n∑_i=1^2^n(tr√(√(ρ_i)√(ρ^'_i)√(ρ_i)))^2≥0.9, 11

where ρ and ρ^' represent the density matrices of the ideal and predicted output states, respectively. It is important to note that we solely rely on fidelity as a metric for comparing PT_VQC to tensor networks and QDNN. The former cost function definitions remain consistent for the PT_VQC and U-VQSVD algorithms. Afterwards, we determine the minimum input state requirements for each algorithm ensuring the proper minimization of the cost function. To ensure the robustness of our results, we assess these requirements across 5 different seeds and calculate their standard deviations.















 











For variational tensor networks <cit.>, the cost function relies on minimizing the Kullback-Leibler divergence <cit.>. The strategy behind this algorithm is to construct the MPO representation of U, assuming structures with low entanglement degrees. During each training step, we evaluate the fidelities between the predicted and actual state vectors. Employing a batch size of 10 samples, we utilize Symmetric Informationally Complete Positive Operator-Valued Measures <cit.> (SIC POVMs), producing 4^n state initializations.

For QDNN <cit.>, the cost function is defined by the fidelity between the QDNN output  ρ_x^out and the correct output ϕ_x^out averaged over the training data size N


    C_QDNN=1/N∑_x=1^N ρ_x^outϕ_x^out12.

To meet the criterion of achieving a fidelity of 0.9 across 5 consecutive seeds, we utilize 3^n initial states for the QDNN scheme. 



 §.§ U-VQSVD algorithm










In this section, we introduce the U-VQSVD algorithm, drawing inspiration from the method for singular value decomposition of a unitary U delivered by  <cit.>. Similar to the unitary process tomography discussed in Section <ref>, the architecture of U_VQC(θ̂) follows that of PT_VQC, with the only change being the sequence of single-qubit gate chains, now arranged as RY-RX-RY as depicted in Fig. <ref>.

Using the U-VQSVD scheme, we propose a new method for process tomography for unknown general channels assuming that the corresponding eigenstates can be learned up to a phase. As a demonstration of the U-VQSVD performance we conduct a forgery attempt on a PE-QPUF <cit.>, with its architecture depicted in Fig. <ref>. The PE-QPUF is constructed with a Haar-random unitary U which, in the user verification stage, probabilistically yields a classical output k' close to the one generated during generation, k <cit.>. Both stages, generation and verification, are constructed using the quantum phase estimation algorithm. Consequently, the architecture comprises two subsets of qubits: ancillary and target qubits. Fig. <ref> illustrates the circuit responsible for generating the initial classical output k with respective quantum state |ψ_k⟩ to be acquired by the user. Conversely, during verification, the post-measurement state obtained during the generation serves as the input for the target system. Typically, the initialization state is not an eigenstate of U due to its Haar-random nature.



 §.§ U-VQSVD attack on PE-QPUF

We outline the steps employed by U-VQSVD to numerically execute an impersonation attack on the PE-QPUF authentication protocol:

 







  
  * We start by preparing 2^n states completing the computational basis {|i⟩}_i=0^2^n-1 and evolve each of them via the VQC UVQC, using the same methodology as for the PT_VQC algorithm in Section <ref>.
  
  * The cost function, instead of being defined with two independent VQCs, as in the work of <cit.>, is just defined with one
  
  
    C(θ̂)=1/2^n∑ _i=0^2^n-1 1-  UVQC(θ̂)^†UUVQC(θ̂)i. 13

    Using the modified definition in Eq. <ref>, the cost of the gradient computation complexity is reduced. This is achieved as the objective has changed, from ideally reconstructing the singular vectors of U, to learning them up to a global phase. 
  
    
  * The parameters θ̂ of the VQC are updated using the Adam optimizer. Consequently, we obtain {|ϕ_i⟩}_i=0^2^n - 1 as the estimated singular vectors of U, where |ϕ_i⟩ = U_VQC(θ̂)|i⟩ and e^iϕ_i =U_VQC(θ̂)^†UU_VQC(θ̂)i is  the i-th estimated  singular value.

  
  * Once the singular values and vectors of U have been learned, the attack occurs during the verification stage. The objective of the attacker is to impersonate the user by sending the learned singular state |ϕ_i⟩, corresponding to the complex phase ϕ_i  which minimizes the quantity |ϕ_i/2 π - k/2^a|.


We assess the efficacy of this novel attack by comparing its capabilities to those of an attacker utilizing random states during the verification procedure, as well as to those of a trusted party undergoing verification.



§ RESULTS
















 §.§ PT-VQC algorithm



In Fig. <ref>, we compare the performance of the PT_VQC algorithm with QDNN, both with (blue) and without (green) hidden layers <cit.>, as well as tensor networks schemes <cit.>. For a scenario involving 2 qubits, with a fixed number of 60 iterations and a target fidelity of 0.9, we observe a faster convergence behavior in our scheme compared to the other two methods. It is worth noting that the attenuated oscillatory behavior observed for PT_VQC in Fig. <ref> arises from the minimization problem being formulated with respect to the cost function defined in Eq.  <ref>, which is not a monotonic function of the average fidelity.

The optimal depth of the VQC for a varying amount of qubits is presented in Fig. <ref> (left).
To model the relationship between the optimal depth and the number of qubits, we conducted an exponential fit of the form D_opt(n) = ae^bn + c, considering the standard deviation of the optimal depth, which was calculated for 5 different Haar random unitaries U. In Fig. <ref> (right), we observe the similarity, as defined in Appendix <ref>, from 2 to 5 qubits over varying iteration numbers for 5 different Haar-random unitaries U. The performance in Fig. <ref> (right) exhibits a convergent behavior, where the amplitude decreases with an increase in the number of qubits. For 5 qubits, only the first 9 iterations were generated due to high processing times. Nevertheless, Fig. <ref> (left) enables us to extrapolate the optimal depth of the VQC for larger qubit sizes.









 §.§ U-VQSVD algorithm



Following the attack presented in Section <ref>, which relies on properly learning the singular values and vectors of the target unitary U of the PE-QPUF, we aim to forge the identity of 10^2 different users. We sample 10^2 Haar-random unitaries for system sizes ranging from 1 to 4 qubits. For each U, we choose 5 different ancilla sizes a ranging from 2 to 6 qubits, resulting in a total of 4 × 5 = 20 PE-QPUFs.

In each scenario, we conduct 25 × 2 ^a forgeries and compute the average discrepancy between the generation and verification outcomes. The data points depicted in Fig. <ref> represent the mean of this average across the entire set of 10^2 different users, with the error bars indicating the fluctuations observed in the comparison across multiple users. In Fig. <ref> (left), we observe the mean deviation of the U-VQSVD attack, contrasting it with Fig. <ref> (right), where an attack with random initializations |Ψ_k⟩ is performed during the verification. Additionally, in Fig. <ref> (middle), we compare it to the reference mean deviation, where a trusted party performs the authentication.


 

In Fig. <ref> we validate the effectiveness of the U-VQSVD targeting unknown quantum unitary evolutions. In other words, it demonstrates how an attack using our U-VQSVD algorithm, directed towards the previously defined PE-QPUF, outperforms an uninformed attack by a factor of 2 to 5 depending on the t and a sizes. It is important to note that this does not undermine the security notions established in <cit.>. In fact, the trusted party-server interaction demonstrates better behavior compared to our forgery attempts. For larger sizes t and a, for which security is proven to hold, our attack would already fail under its own assumptions, implying that the resources required for such a scheme scale exponentially.











§ CONCLUSIONS




    This work non-trivially modifies two existent proposals found  in the state of the art of  VQC for process tomography, enhancing their performances and discussing their actual overhead in a comprehensible manner by introducing the concept of optimal depth. The mentioned modifications allow us for the creation of what we denote as the PT_VQC and U-VQSVD algorithms. 
    
    Within our first scheme, the learning of unknown unitary operators is tackled and the overhead of such learning is considerably reduced with respect to the proposal in <cit.>: 4^n required initializations are brought to only 2^n and the amount of qubits in each run is halved. Remarkably, such reduction of the scheme resources do not prevent us from accurately reconstructing Haar-random targets, with higher convergence rate than the one obtained using QDNN or tensor network schemes targeting lowly entangling evolutions.
    
    Regarding the second algorithm we deliver, U-VQSVD, we manage to reduce the learning complexity for singular value decomposition (SVD) displayed in <cit.> by relaxing its objective: instead of asking for an exact SVD of unitaries, we seek a decomposition where singular vectors are learned up to a global phase. Such relaxation leaves us with a functional algorithm within the realm of Quantum Physics, where  global phases of states do not posses a physical meaning. Supporting this statement, we manage to outperform a random attack against the PE-QPUF defined in <cit.> by explicitely exploiting the U-VQSVD algorithm.

    It is worth noting that all the VQC employed have successfully provided solutions to the problem following a gradient-based minimization of the cost function. We conjecture that the optimization problems posed by our cost functions, defined in Eq. <ref>, Eq. <ref>, and the parameterized ansätze are not ill-posed. In other words, they do not possess undesirable local minima. The proof supporting this conjecture may pave the way for a new theoretical research direction in VQC optimization.





In this study, we introduce two innovative process tomography techniques: PT_VQC and U-VQSVD.

PT_VQC (Section. <ref>) compared to <cit.> significantly reduces the required number of state initializations to perform process tomography from 4^n to 2^n and halves the necessary number of qubits for each run. This enhancement results in improved algorithm processing time and increased unitary reconstruction accuracy. PT_VQC surpasses both QDNN and tensor networks (MPO), as demonstrated in Fig. <ref>. Unlike tensor networks utilizing MPOs, which assume low entanglement structures, PT_VQC performs well regardless of the structure of U.

U-VQSVD (Section. <ref>) presents a novel process tomography scheme designed for unknown general channels. It assumes that the corresponding eigenstates can be learned up to a global phase and is based on the variational quantum singular value decomposition (VQSVD) architecture proposed in <cit.>. To evaluate the efficacy of the U-VQSVD algorithm, we conduct an impersonation attack on a PE-QPUF <cit.>. We compare the capabilities of an uninformed attacker, employing random states, with those of an attacker using U-VQSVD to generate input states, alongside a trusted party. U-VQSVD outperforms the uninformed attack by a factor of 2 to 5, depending on the ancilla and target sizes of the PE-QPUF.

It is worth noting that all the VQC employed have successfully provided solutions to the problem following a gradient-based minimization of the cost function. We conjecture that the optimization problems posed by our cost functions, defined in Eq. <ref>, Eq. <ref>, and the parameterized ansätze are not ill-posed. In other words, they do not possess undesirable local minima. The proof supporting this conjecture may pave the way for a new theoretical research direction in VQC optimization.










§ SIMILARITY CORRELATION TO THE COST FUNCTION



 
The complementary of the similarity  between the target and learned  operators  is  a monotonically increasing function of the cost defined in  Eq. <ref>. 
The vanishing of the cost function  defined in  Eq. <ref> implies a correct learning of the targeted operator.





The complementary of the similarity, S(A,B), between two operators A and B is defined  via the Frobenius norm <cit.> as


    1-S(A,B)=A-B_F/A_F, 14


where, if x_ij are the entries of an operator X in an orthonormal basis {|i⟩}_i=0^2^n-1, then


    X_F= √(∑_i^2^n-1∑_j^2^n-1x_ij^2).15


For the unitary  operators U and U(θ̂)_VQC, we have


    1-S(U,U(θ̂)VQC) ∝^+  (U-U(θ̂)VQC)_F, 16


where ∝^+ stands for "proportional via a positive constant".

If we define ⟨k|(U-U(θ̂)VQC)|i⟩≡α_k^i, then


    1- S(U,U(θ̂)VQC) ∝^+ √(∑_i=0^2^n-1∑_k=0^2^n-1 |α^i_k|^2)17        ,



    = √(∑_i=0^2^n-1| |(U-U(θ̂)VQC) ) |i⟩| |^2),     18


leading to


    1- S(U,U(θ̂)VQC) ∝^+ √(2 ∑_i=0^2^n-1   1 - {⟨i|U^† U(θ̂)_VQC|i⟩}), 19


to finally obtain


    1- S(U,U(θ̂)VQC) ∝^+ √(f(θ̂)). 20


The fact that g(x)=√(x) is a monotonically increasing function added to the fact that √(0) = 0, concludes the proof.




§ COMPUTATION OF THE REAL AND IMAGINARY PART OF AN OVERLAP BETWEEN QUANTUM STATES


Considering a Hilbert space of dimension 2^n with |ψ⟩, |ϕ⟩ ∈ ℋ^⊗ n. The inner product ⟨ϕ||ψ⟩ gathers the real and imaginary parts that we want to obtain, with the assumption of owning several copies of both |ψ⟩ and |ϕ⟩. Let V and W be unitary operators such that V|0⟩^⊗ n = |ψ⟩ and  W|0⟩^⊗ n = |ϕ⟩.



  
*Real part 
We can calculate the real part by carrying out the following circuit on (n+1)-qubits register


    (H⊗ 11)(C_1W    )(C_0V  ) (H ⊗ 11)|0⟩ancilla⊗|0⟩^⊗ n, 21



hence, obtaining


    {⟨ϕ||ψ⟩} = 2 p_0-1, 22


where the subscript in C_x stands for the choice of the state to be controlled and p_0 is the probability of obtaining the classical outcome "0" when measuring the ancillary qubit.



  
*Imaginary part 
We can also calculate the imaginary part by carrying out the following circuit on (n+1)-qubits register



    (H ⊗ 11)(C_1W    )(C_0V  ) (P(3/2π)H⊗ 11)|0⟩_ancilla⊗|0⟩^⊗ n,23


hence, obtaining


    {⟨ϕ||ψ⟩} = 2p_0 -1. 24



With the real and imaginary parts defined, the attacker can calculate |ϕ⟩= W(θ̂)̂|i⟩ and |ψ⟩ = UW(θ̂)̂|i⟩, in order to obtain the singular value λ_i corresponding to the i-th singular vector.



§ PSEUDO-CODE FOR PT_VQC


The routines for the PT_VQC algorithm are described below. Our objective is to achieve a cost function C(θ̂) ≥ 0.10 for 5 consecutive Haar-random unitaries (with a threshold of 5). The same pseudo-code applies for U-VQSVD, but instead of the 4-term shift rule, we use the 2-term shift rule <cit.>.




§ CODE AVAILABILITY

The github repository can be found at: https://github.com/terrordayvg/PT_VQC-Tomographyhttps://t.ly/REETC.












 




















































§ ACKNOWLEDGEMENT

Christian Deppe and Roberto Ferrara were supported by the German Federal Ministry of Education and Research (BMBF), Grant 16KIS1005.
Christian Deppe acknowledge the financial support by the Federal Ministry of Education and Research of Germany in the programme of ”Souverän. Digital. Vernetzt”. Joint project 6G-life, project identification number: 16KISK002.
Vladlen Galetsky, Soham Ghosh and Christian Deppe were supported by the BMBF Project 16KISQ038.
Christian Deppe and Roberto Ferrara are supported by the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus.

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