Maximum-Likelihood Quantum State Tomography by Soft-Bayes
Anonymous Author(s)
AffiliationAddress
Abstract
Quantum state tomography (QST), the task of estimating an unknown quantum stategiven measurement outcomes, is essential to building reliable quantum computingdevices.
Whereas computing the maximum-likelihood (ML) estimate correspondsto solving a finite-sum convex optimization problem, the objective function is notsmooth nor Lipschitz, so most existing convex optimization methods lack samplecomplexity guarantees; moreover, both the sample size and dimension grow exponentially with the number of qubits in a QST experiment, so a desired algorithmshould be highly scalable with respect to the dimension and sample size, just likestochastic gradient descent.
In this paper, we propose a stochastic first-order algorithm that computes an ε -approximate ML estimate in O (( D log D ) /ε 2 )
iterationswith O
( D 3 ) per-iteration time complexity, where D denotes the dimension of theunknown quantum state and ε denotes the optimization error.
Our algorithm is anextension of Soft-Bayes to the quantum setup.
Introduction
Quantum state tomography (QST), the task of estimating an unknown quantum state givenmeasurement outcomes, is essential to building reliable quantum computing devices [49].
The statesof the quantum bits (qubits) prepared by an experimental apparatus are estimated, in order to checkthe correctness of the apparatus and, if needed, determine how to calibrate it.
Moreover, quantumprocess tomography, the task of estimating an unknown quantum channel, can also be cast as a QSTproblem [6].
There are various approaches to QST, such as trace regression [46, 26, 38, 23, 60, 61],maximum-likelihood (ML) estimation [31, 32, 11], Bayesian estimation [9, 10], and recentlyproposed deep learning-based methods [3, 50].
Among existing approaches, the ML approach hasbeen widely adopted for its relatively low estimation error in practice and asymptotic statisticalguarantees in theory [32, 52].
Computing the ML estimator amounts to solving an optimization problem.
Whereas the optimizationproblem is convex, standard convex optimization methods are not directly applicable.
It is easilychecked that the negative log-likelihood function in ML QST is neither Lipschitz nor smooth,violating standard assumptions in optimization literature [36, 22].
Hence, for example, even whethervanilla gradient descent converges for QST is unclear.
The negative log-likelihood function is indeedself-concordant, so Newton’s method is readily applicable [42].
Nevertheless, the dimension of aquantum state grows exponentially with the number of qubits; the Hessian computations in Newton’smethod are computationally too expensive when the dimension is high.
There are a few first-orderconvex optimization algorithms provably converging for ML QST, such as diluted RρR [57, 25],
SCOPT [54] 1 , NoLips [8], the Frank-Wolfe method [45, 22, 62, 13], and entropic mirror descent with line search [36].
These are all batch methods.
Their per-iteration time complexities are at least linearin the sample size.
Unfortunately, to estimate a quantum state, the sample size must be exponential inthe number of qubits in general [44, 28].
Regarding the high dimension and sample size issues in ML QST, it is desirable to, like how we handlethe same issues in modern machine learning applications, develop a stochastic first-order optimizationmethod for ML QST.
The stochastic quasi-Newton method for self-concordant minimization of Zhouet al.
[63] seems to apply.
Nevertheless, its step size selection rule involves Hessian computations,computationally expensive in ML QST; moreover, its analysis assumes a bounded Hessian, whichdoes not hold in ML QST.
The stochastic mirror-prox and stochastic primal-dual hybrid gradientmethods were considered for problems very similar to ML QST [4, 15, 30].
However, their analysesassume either a bounded dual domain or Lipschitzness; both are violated in ML QST.
In this paper, we propose a stochastic first-order algorithm for ML QST.
We design the algorithm by anonline learning argument.
Consider an online convex optimization problem, where the loss functionin each round corresponds to the negative log-likelihood function corresponding to one data point in
ML QST.
Interestingly, this online convex optimization problem is exactly the quantum analogueof online portfolio selection, a celebrated online learning problem [17, 18].
Since the ML approachaims to minimize the empirical average of the negative log-likelihood, once we “quantumize” anyexisting first-order online portfolio selection algorithm that is no-regret and apply an online-to-batchconversion [14, 20], the resulting algorithm becomes a stochastic first-order algorithm for ML QST.
We refer the reader to Section 2 for an introduction of relevant concepts.
The algorithm we choose to “quantumize” is Soft-Bayes [48].
There are two reasons.
First, theper-round time complexity of Soft-Bayes is linear in the ambient dimension, arguably the lowestone can expect; second, Soft-Bayes has a curious connection with expectation maximization [40, 41](see Section 5).
We call the resulting algorithm Stochastic Q-Soft-Bayes.
Stochastic Q-Soft-Bayesprocesses one randomly chosen data point in each iteration.
Suppose the quantum state to be estimatedis represented by a D -by- D density matrix.
The per-iteration time complexity of Stochastic Q-SoftBayes is O
( D 3 ) , independent of the sample size.
The expected optimization error of Stochastic QSoft-Bayes converges to zero at a O ( (cid:112) (1 /T )
D log D ) rate, where T denotes the number of iterations.
The main technical difficulty lies in figuring out an appropriate quantum extension of Soft-Bayesthat coincides with Soft-Bayes when all matrices involved share the same eigenspace and allowsfor a regret analysis.
This is challenging because for any given “non-quantum” expression, one canimmediately find many candidates for its quantum extension, but only a few or one of them inheritthe desired theoretical properties of their “non-quantum” counterpart; see, e.g., the discussion in[59, Chapter 11] for extending information theoretic quantities to the quantum case.
Similar to thequantum extension of exponentiated gradient update by Tsuda et al.
[55], the quantum extension wefind reveals the complicated mathematical structure of Soft-Bayes hidden in the “non-quantum” setup.
Instead of empirically beating state of the arts, our aim is to give the first provably fast stochastic firstorder algorithm for ML QST.
According to the comparison in Section 3.3, Stochastic Q-Soft-Bayesis actually competitive in time complexity in comparison to existing batch algorithms.
Unfortunately,
Section 4 shows that the empirical speed of Stochastic Q-Soft-Bayes is not satisfactory to practitioners.
We discuss the possibility of developing faster stochastic first-order methods in Section 5.
1.1 Related work
A textbook approach to quantum state tomography is to approximate the problem as a trace regressionproblem [43] and compute the corresponding least-squares estimate or directly minimize the expectedsquare loss, sometimes with regularization [46, 26, 23, 61, 60].
Since minimizing the square loss isarguably the most standard problem in optimization and machine learning, many existing algorithmsapply.
Youssry et al.
[61] proved the convergence of stochastic entropic mirror descent.
Yang et al.[60] showed that several standard online learning algorithms are no-regret for the correspondingonline trace regression problem.
Notice that both papers do not consider the ML formulation.
Quantum state tomography schemes optimal or nearly optimal in sample complexity were proposed in[27, 28, 44].
The schemes in [44] and [28] require entangled measurements, challenging to implement.
Whereas the scheme studied in [27] allows independent measurements as in ML QST, the numericalresult in [27, Figure 1] shows that the ML approach achieves a smaller estimation error empirically.
A problem closely related to quantum state tomography is shadow tomography, in which one isnot interested in recovering the quantum state but estimating the probability distributions of itsmeasurement outcomes [1].
Aaronson et al.
showed that shadow tomography can be done in an onlinefashion, via follow the regularized leader with the von Neumann entropy [2].
We emphasize thatshadow tomography is fundamentally different from quantum state tomography.
Indeed, Aaronsonshowed shadow tomography is strictly easier than state tomography, in the sense that the formerrequires much less samples than the latter [1].
Another closely related problem is the quantum versionof individual sequence prediction considered by Koolen et al.
The loss function studied in [35]is the trace-log loss, instead of the log-trace loss we consider, as discussed in Section 4 of their paper.
Our algorithm is developed via “quantumizing” an online portfolio selection algorithm.
Onlineportfolio selection is a classic online learning problem.
It is known that the optimal regret of onlineportfolio selection is O ( D log T ) , where D denotes the ambient dimension and T denotes thenumber of rounds, and is achieved by Universal Portfolio Selection (UPS) [17, 18].
However, UPSis computationally too expensive to be practical [33].
There are several algorithms that try to balancebetween the regret and computational complexity, but none of them is optimal in both aspects [39, 56].
Soft-Bayes strikes a balance with a O ( D ) per-round time complexity and O ( √ TD log D ) regret.
Recently, Zimmert et al.
[64] “quantumized” another online portfolio selection algorithm, called
BISONS, to solve the game of online quantum state tomography described in Section 3.1 2 .
By anonline-to-batch conversion, their algorithm yields a stochastic algorithm for ML QST.
The resultingalgorithm achieves a better iteration complexity than Stochastic Q-Soft-Bayes; nevertheless, eachiteration of it requires solving a self-concordant convex program by, e.g., Newton’s method, resultingin a high time complexity incomparable to that of Stochastic Q-Soft-Bayes.
In the words of Zimmertet al.
[64], both their and our algorithms are on the state-of-the-art efficiency-regret frontier.
Notations
We write R + for the set of non-negative real numbers and R ++ the set of strictly positive real numbers.
Let J ∈ N .
We write [ J ] for the set { 1 , . . .
, J } .
Let M be a matrix.
We write M H for its Hermitian(conjugate transpose) and tr( M ) for its trace.
Let H be a Hermitian matrx; we write its spectraldecomposition as H =
(cid:80) d λ d P d , where λ d are the eigenvalues and P d are projections onto theassociated eigenspaces.
Let f be a real-valued function whose domain contains { λ d } .
Then, f ( H )is defined as the matrix (cid:80) d f ( λ d ) P d .
Let A and B be two matrices.
We write A ≥ B if and only if
A − B is positive semi-definite.
Let E be an event and ξ be a random variable following a probabilitydistribution P .
We write P ( E ) for the probability of the event and E P [ ξ ] for the expectation of ξ .
Wesometimes omit the subscript P and write E [ ξ ] when there is no ambiguity.
Preliminaries
In this section, we provide the background knowledge relevant to this paper.
We review the problemformulations of ML QST, log-optimal portfolio, and online portfolio selection and discuss theconnections among them.
Then, we review Soft-Bayes, the particular online portfolio selectionalgorithm we focus on.
2.1 Maximum-Likelihood Quantum State Tomography
In the mathematical formulation of quantum mechanics, a quantum state corresponds to a densitymatrix , a Hermitian positive semi-definite complex matrix of unit trace.
Let the dimension of thedensity matrix be D ∈ N .
If there are q qubits, then D = 2 q .
We denote by D the set of densitymatrices in C D × D , i.e.,
[Equation]
[Equation]
A measurement setup corresponds to a positive operator-valued measure (POVM) , a set of Hermitianpositive semi-definite complex matrices summing to the identity.
Let ρ ∈ D and { M 1 , . .
, M J } ⊂
C D × D be a POVM.
The measurement outcome is a random variable η taking values in [ J ] such that
P
[Equation]
The ML estimation approach seeks the quantum state that maximizes the probability of observing themeasured data.
Let ρ ♮ ∈
D be the density matrix to be estimated.
In a standard QST experiment, weconstruct N independent copies of ρ ♮ and measure the copies independently with possibly different
POVMs.
It is easily checked that the ML estimator is of the form
[Equation]
where each A n is an element of the POVM for the n -th measurement.
We call { A 1 , . . .
, A N } thedata-set .
We equivalently write the ML estimator as
[Equation]
Obviously, (1) is a convex optimization problem.
If the matrix A n ′ is not full-rank for some n ′ ∈ [ N ](as in the cases with the Pauli measurement [37] and Pauli basis measurement [51, 53]), then tr( A n ′ ρ )can be arbitrarily close to zero on D and hence the k -th-order derivative of the objective function f isunbounded for all k ∈ N .
Let A be a random matrix following the empirical distribution ˆ P N on the data-set { A 1 , . . .
, A N } .
If the matrices A n are all different, then ˆ P N is simply the uniform distribution on the data-set{ A 1 , . . .
, A N } .
Then, we can write the objective function in (1) as an expectation
[Equation]
This observation connects ML QST with the problem of computing the log-optimal portfolio.
2.2 Log-optimal Portfolio
[Figure] Interestingly, the optimization problem (1) is exactly a quantum extension of the problem of computing the log-optimal portfolio (aka the Kelly criterion), an asymptotically optimal strategy forlong-term investment [5, 12, 34].
Consider multi-round investment in a market.
Suppose there are Dinvestment alternatives.
For the t -th round, we list the return rates of the investment alternatives inthat round as a random vector a t ∈ R D + .
Before each round starts, an investor needs to determinethe portfolio for the round given the past return rates.
Denote by P t +1 the probability distribution ofa t
+1 conditional on the history ( a 1 , . .
. , a t ) .
The log-optimal portfolio w ⋆ t +1 for the ( t + 1) -th roundis given by the stochastic optimization problem:
[Equation]
where ∆ denotes the probability simplex in R D , the set of entry-wise non-negative vectors whoseentries sum to one.
Then, the investor distributes the wealth to the investment alternatives followingthe ratios specified by w ⋆t +1 .
We now discuss the correspondence between ML QST and log-optimal portfolio.
The set D is indeeda quantum extension of the probability simplex ∆ , in the sense that a Hertimian matrix is a densitymatrix if and only if its vector of eigenvalues lies in the probability simplex.
The objective functionsin (1) and (4) are both expectations of the logarithm of linear functions.
Indeed, it is easily checkedthat if the matrices involved in (1) share the same eigenbasis, then the non-commutativity issue in thequantum setup vanishes and (1) coincides with (4).
Though the correspondence is obvious given thetwo problem formulations, it seems that this correspondence has not been discussed in the literature.
2.3 Online Portfolio Selection
Online portfolio selection may be viewed as a probability-free version of log-optimal portfolio [17].
Online portfolio selection is a multi-round game between two players, say I NVESTOR and M ARKET .
Suppose the game consists of T rounds.
In the t -th round of the game, first, I NVESTOR announces aportfolio w t ∈ ∆ ; then, M ARKET announces the return rates of all investment alternatives for the t -thround in a vector a t ∈ R D + ; finally, I NVESTOR suffers a loss of value − log ⟨ a t , w t ⟩ .
The goal of I NVESTOR is to achieve a low regret against all possible strategies of M ARKET .
The regret is defined as
[Equation]
where the supremum is over all possible strategies of M ARKET to determine ( a t ) 1 ≤ t ≤ T .
We sayan algorithm for I NVESTOR to determine the portfolios is no-regret if it achieves R T
= o ( T ) .
If we can sample from the conditional probability distribution P t +1 specified in the previous subsection, then we can transform a no-regret online portfolio selection algorithm to an algorithm thatapproximately computes the log-optimal portfolio.
The following is an immediate consequence ofthe online-to-batch conversion [14, 47].
Proposition 2.1.
Suppose in the online portfolio selection game, the vectors a t are all independentand identically distributed (i.i.d.)
random vectors following the probability distribution P t +1 inthe previoius sub-section.
Let ( w t ) t ∈
N be the sequence of iterates generated by an algorithm for
I NVESTOR of regret R T .
Then, for any T ∈ N ,
[Equation]
T
T .
Recall that φ is the conditional expectation of the log-linear loss in (5) .
If I NVESTOR adopts a no-regret algorithm, then the expected optimization error vanishes as T → ∞ .
2.4 Soft-Bayes
There are various existing algorithms for online portfolio selection.
Among these algorithms, we areparticularly interested in Soft-Bayes [48].
The per-iteration time complexity of Soft-Bayes is linearin D , arguably the lowest one can expect.
This is a desirable feature for ML QST, as the dimensionof the density matrix grows exponentially with the number of qubits.
The iteration rule of Soft-Bayes is as follows.
• Initialize at w 1 =
(1 /D, . . .
, 1 /D ) ∈ ∆ (the uniform distribution).
• For each t ∈ N , compute
[Equation]
for some properly chosen learning rate η ∈ [0 , 1] , where ◦ denotes the entry-wise product.
Soft-Bayes has the following regret guarantee.
Theorem 2.2 ([48]) .
After T rounds in online portfolio selection, the regret of Soft-Bayes with
[Equation]
is at most 2 √ TD log D + log D .
Online Maximum-Likelihood Quantum State Tomography by Q-Soft-Bayes
Following the discussion in Section 1, we first propose a game of online quantum state tomographyas a quantum extension of online portfolio selection.
Then, we “quantumize” Soft-Bayes to derivea no-regret algorithm for the game and analyse its regret.
Finally, we adopt the online-to-batchconversion and bound the expected optimization error of the resulting algorithm.
3.1 Game of Online Quantum State Tomography
We propose the following game of online quantum state tomography as a quantum extension ofonline portfolio selection.
Online quantum state tomography is a multi-round game between twoplayers, say P HYSICIST and E NVIRONMENT .
Suppose there are in total T rounds.
In the t -th round,first, P HYSICIST announces a density matrix ρ t ∈ D ; then, E NVIRONMENT announces a Hermitianpositive semi-definite matrix A t ≥ 0 ; finally, P HYSICIST suffers for a loss of value − log tr
( A t ρ t ) .
The regret in this game is given by
[Equation]
where the supremum is over all possible strategies of P HYSICIST to generate the sequence ( A t ) 1 ≤ t ≤ T .
The connection with online portfolio selection is obvious and similar to that between ML QST andthe log-optimal portfolio: The vector of eigenvalues of a density matrix lies in the probability simplex∆ ; the Hermitian matrices A t and the positive semi-definiteness condition correspond to the vectorsa t in online portfolio selection and their non-negativity condition, respectively; the losses in the twogames are both logarithms of linear functions; the regrets in the two games are defined exactly in thesame manner.
When all the matrices involved in the game of online quantum state tomography sharethe same eigenbasis, we recover the game of online portfolio selection.
3.2 Q-Soft-Bayes and Stochastic Q-Soft-Bayes
We propose the following Q-Soft-Bayes algorithm as a quantum extension of Soft-Bayes.
• Initialize at ρ 1 = W 1 = I/D .
• For each t ∈ N , compute
[Equation]
for some properly chosen learning rate η ∈ [0 , 1] .
Remark 3.1 .
Recently, we learned that Q-Soft-Bayes may be interpreted using the commutativematrix product by Warmuth and Kuzmin [58].
It is currently unclear to us whether this interpretationprovides any insight.
If we were able to cancel the exponential and logarithms in Q-Soft-Bayes, then we recover Soft-Bayes;however, due to the non-commutativity issue, such cancellation is illegal in general.
In comparison to
Soft-Bayes, Q-Soft-Bayes has an additional normalization step to ensure its outputs are of unit trace.
We prove the following in Section B.
Proposition 3.2.
It holds that tr( W t ) ≤ 1 for all t .
Numerical experiments show that the equality does not always hold, so the normalization step isnecessary.
Recall that Soft-Bayes does not need the normalization step (see Section 2.4).
In Appendix C, we prove the following regret bound for Q-Soft-Bayes, showing that it inherits theregret bound of Soft-Bayes.
Theorem 3.3.
The regret of Q-Soft-Bayes with the learning rate η given in (7) is at most2 √ TD log D + log D .
Remark 3.4 .
One might wonder why D is not replaced by D 2 in the quantum case.
This is becausein our extension, the analogue of a D -dimensional vector in the quantum case is the D -dimensionalvector of eigenvalues of a D -by- D Hermitian matrix, instead of the D -dimensional vector obtainedby vectorizing a √ D -by- √ D matrix.
Similar coincidence of regret bounds can be observed in, forexample, the matrix version of exponentiated gradient update [55, 7] and the quantum individualsequence prediction algorithms of Koolen et al.
The standard online-to-batch conversion argument can also be applied to solving ML QST by
Q-Soft-Bayes.
Recall for ML QST, our aim is to solve the stochastic optimization problem
[Equation]
where A is a random matrix following the empirical probability distribution ˆ P N on the data-set{ A 1 , . .
A N } (see Section 2.1).
We propose the following stochastic optimization algorithm, whichwe call Stochastic Q-Soft-Bayes, to solve ML QST.
• Initialize Q-Soft-Bayes with ρ 1 = W 1 =
I/D .
• In the t -th iteration of Stochastic Q-Soft-Bayes, do the following.
Output the t -th output ρ t of Q-Soft-Bayes.
Sample a random matrix B t ∈ { A 1 , . . .
, A N } following the empirical probabilitydistribution ˆ P N on the data-set, independent of the past.
Let E NVIRONMENT in the online QST game announce the matrix B t .
Similarly as for Proposition 2.1, the standard online-to-batch conversion argument provides thefollowing convergence guarantee of Stochastic-Q-Soft-Bayes.
Proposition 3.5.
Let ( ρ t ) t ∈
N be the sequence of iterates generated by Stochastic Soft-Bayes.
Then,for any T ∈ N , it holds that
[Equation]
where ρ T :=
( ρ 1 + · · · + ρ T ) /T
and the expectation is with respect to the randomness in B t of
Stochastic Soft-Bayes.
Recall f is the objective function in ML QST as defined in (2) or (3) (the twodefinitions are equivalent).
Therefore, Stochastic-Q-Soft-Bayes outputs an approximate ML estimator of expected optimizationerror smaller than ε in O (( D log D ) /ε 2 ) iterations.
Each iteration of Stochastic-Q-Soft-Bayesrequires computing a matrix exponential and two matrix logarithms.
The overall time complexity ishence O (( D 4 log D ) /ε 2 ) .
3.3 Theoretical Comparison with Existing Batch Algorithms
Let us compare the time complexities of Stochastic Q-Soft-Bayes and existing algorithms discussedin Section 1.
The iteration complexities of existing algorithms are mostly unknown or vague intheir dependence on the problem parameters.
Diluted RρR and entropic mirror descent with linesearch do not have non-asymptotic analysis results [57, 25, 36]; SCOPT only has a local linearrate guarantee [54]; Adaptive Frank-Wolfe and Monotonous Frank-Wolfe have O ( ε − 1 ) iterationcomplexities with unclear dependence on the dimension and sample size, as their error boundsinvolve local smoothness parameters that are hard to evaluate [13, 22].
A finer analysis of Adaptive Frank-Wolfe by Zhao and Freund [62] shows that its iteration complexity is O ( ε − 1 N ) and henceits time complexity is O (cid:0) ε − 1 ( N 2 D 2 + Nτ ) (cid:1) , where the symbol τ denotes the time of computingthe local norm defined by the Hessian, for which we do not know an efficient implementation.
Incomparison, the complexities of Stochastic Q-Soft-Bayes is very clear: O ( ε − 2 D log D ) iterationcomplexity and hence O ( ε − 2 D 4 log D ) time complexity.
The time complexity of Stochastic Q-SoftBayes becomes competitive with Adaptive Frank-Wolfe if N ≫ D (cid:112) (1 /ε ) log D , ignoring the timeof computing the local norms .
It is known that N = Ω( D 2 /δ ) is required to obtain an estimate offidelity δ (see the next section for the definition of fidelity) [44].
The algorithm by Zimmert et al.[64] has a ˜ O
( D 3 /ε ) iteration complexity and O ( D 6 ) per-iteration time complexity ignoring thedependence on other parameters, due to the use of Newton’s method; the overall time complexity hasa much higher dependence on the dimension than Adaptive Frank-Wolfe and Stochastic Q-Soft-Bayes.
We conclude that the time complexity of Stochastic Q-Soft-Bayes is competitive in comparison toexisting algorithms.
Numerical Results
As discussed above, the proposed algorithm is competitive in time complexity. The empiricalconvergence speed of Stochastic Q-Soft-Bayes, unfortunately, is slower than existing batch methods.
We compare its empirical speed with two batch methods, the RρR method [40, 41] and Monotonous Frank-Wolfe [13], on a synthetic data-set in Figure 1 and Figure 2.
We have mentioned several batchmethods applicable for ML QST in Section 1.
Among them, we choose RρR for comparison as itis representative in physics literature and empirically fast, though it does not always converge .
Wechoose monotonous Frank-Wolfe for comparison as it avoids computationally expensive Hessiancomputations in step size selection and still converges at a O (1 /t ) rate as other Frank-Wolfe methodsfor self-concordant minimization do [62, 22, 45].
The synthetic data-set is generated basically following the set-up in [29].
The number of qubits qequals 4.
The dimension D then equals 2 q = 16 .
The unknown quantum state to be measured is the
W state.
We randomly generate N = 4 q × 100 = 25600 Pauli observables as in, e.g., [23, 26, 37],each of which corresponds to a POVM of two rank- ( D/ 2) elements.
As there are in total 4 q different
Pauli observables (and hence POVMs), each observable is used about 100 times.
Then, we samplethe N measurement outcomes and formulate the ML estimator following Section 2.1.
The performance measures we consider are optimization errors (in objective function) and fidelityvalues.
As we do not have the exact minimum value of the objective function, we run each algorithmfor 200 epochs and use the smallest function value found by the algorithms, the one found by
Monotonous Frank-Wolfe, as an approximate optimal value.
The approximate optimization errorof an iterate is defined as the difference between the objective function value at the iterate and theapproximate optimal value.
Fidelity is a notion commonly used by physicists to measure how closetwo quantum states are to each other.
For any two density matrices ρ and σ , the fidelity is givenby F ( ρ, σ ) :
= (cid:0) tr (cid:112) √ ρσ √ ρ (cid:1) 2 , which takes values in [0 , 1] .
The fidelity of two quantum statesequals 1, if the two states are exactly the same.
We plot the optimization errors and fidelity valuesversus the number of epochs.
An epoch corresponds to one pass of the whole data-set.
One iterationof Stochastic Q-Soft-Bayes corresponds to 1 /N epoch.
One iteration of RρR and Monotonous Frank-Wolfe corresponds to 1 epoch as both algorithms are batch.
We can observe that Monotonous Frank-Wolfe is the fastest in both figures.
This observation maybe explained by the fact that Frank-Wolfe tends to generate approximately low-rank iterates.
The Wstate corresponds to a rank-1 density matrix, so the ML estimate should be approximately low-rank,matching the structure favoured by Frank-Wolfe.
We can observe that Stochastic Q-Soft-Bayes isfaster than RρR in the low-precision region, but then slower than RρR when the number of epochsbecomes large. This is perhaps due to the sub-optimality of Soft-Bayes, the foundation of Stochastic
Q-Soft-Bayes, with respect to its regret performance in online portfolio selection. We also compare the algorithms in terms of the elapsed time, which can be found in Appendix A.
The codes are provided in the supplementary material.
Discussions
5.1 Can We Find a Faster Stochastic First-Order Algorithms for ML QST?
Our approach to constructing a stochastic first-order algorithm for ML QST conceptually appliesto any no-regret online portfolio selection algorithm.
In this paper, we focus on Soft-Bayes.
Otherexisting online portfolio selection algorithms have much higher per-iteration time complexities, interms of the dependence on the ambient dimension and sample size.
If we adopt any other existingonline portfolio selection algorithm and “quantumize” it to obtain a stochastic algorithm for ML
QST, then the resulting algorithm will scale poorly with the number of qubits.
Developing an onlineportfolio selection algorithm that enjoys both a low regret and low time complexity is still an openproblem [56, 64].
It is still possible to develop another quantum extension of Soft-Bayes that enjoys a lower periteration time complexity.
The per-iteration time complexity issue may be mitigated if we considerother quantum extensions of Soft-Bayes.
For example, if we naïvely replace (8) by W t +1 =( G t W t + W t G t ) / 2 , the resulting algorithm still coincides with Soft-Bayes when all matrices sharethe same eigenbasis, whereas the per-iteration time complexity is reduced to O
( D 2 ) .
Unfortunately,we do not have a non-asymptotic analysis for any other possible quantum extension of Soft-Bayes wecan think of.
The discussion above assumes that we adopt the online-to-batch argument as in this paper.
Anotherway, which we think perhaps more plausible, is to directly consider the stochastic optimizationformulation and develop a stochastic optimization algorithm for ML QST.
5.2 Connection with Expectation Maximization
Finally, let us discuss an interesting connection between Q-Soft-Bayes and expectation maximization(EM).
The RρR algorithm, according to [40, 41], was inspired by the expectation maximization (EM)method for solving optimization problems of the form (4).
Given a full-rank initial iterate ρ 1 ∈ D ,
RρR iterates as
[Equation]
where f is defined in (2).
In comparison, given an entry-wise positive vector w 1 ∈ ∆ , EM for (4)iterates as
[Equation]
It is interesting to notice that even when all matrices involved share the same eigenbasis, RρR isnot equivalent to EM.
Indeed, EM is proved to asymptotically converge to the optimum [16, 19],whereas RρR oscillates on a carefully designed data-set [57].
This suggests that RρR is perhaps nota “natural” quantum extension of EM.
Later, there were variations of RρR that solve the convergenceissue by line search [57, 25], but these variations still do not recover EM.
Notice that the formulation of Soft-Bayes (6) is the convex combination of the previous iterate andthe output of EM .
Therefore, Soft-Bayes, after the online-to-batch conversion, can be interpreted as arelaxed stochastic EM method for computing the log-optimal portfolio.
As Q-Soft-Bayes becomes
Soft-Bayes when all matrices involved share the same eigenbasis, we may claim that Stochastic
Q-Soft-Bayes is also a relaxed stochastic EM method, though its derivation does not have any obviousrelation with the standard derivation of EM [21].
Checklist
all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’scontributions and scope?
[Yes](b) Did you describe the limitations of your work?
[Yes] See Section 4.
Whereas theproposed algorithm is fast in theory, its speed is unfortunately not satisfactory topractitioners.
(c) Did you discuss any potential negative societal impacts of your work?
[No] This is atheory work.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
[Yes]
If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results?
[Yes] Indeed, our algorithm applies to every data-set generated following Section 2.1.
(b) Did you include complete proofs of all theoretical results?
[Yes] See the appendix.3.
If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)? [Yes] The codes are attached.
(b) Did you specify all the training details (e g., data splits, hyperparameters, how they were chosen)?
[Yes] See Section 4.
(c) Did you report error bars (e g., with respect to the random seed after running experi- ments multiple times)?
[No] The comparison is obvious, so we do not feel it necessaryto report error bars.
(d) Did you include the total amount of compute and the type of resources used (e g., typeof GPUs, internal cluster, or cloud provider)?
[Yes] See Section A.
This information is
not necessary for Section 4.
If you are using existing assets (e g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators?
[N/A]
(b) Did you mention the license of the assets?
[N/A]
(c) Did you include any new assets either in the supplemental material or as a URL?
[N/A]
(d) Did you discuss whether and how consent was obtained from people whose data you’re
using/curating?
[N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content?
[N/A]
If you used crowdsourcing or conducted research with human subjects...
[Table]
(a) Did you include the full text of instructions given to participants and screenshots, if
[Table]
applicable?
[N/A]
[Table]
(b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable?
[N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount
spent on participant compensation?
[N/A]