Improved robustness of quantum supremacy for random circuit sampling
aa r X i v : . [ qu a n t - ph ] F e b Fine-grained analysis and improved robustness ofquantum supremacy for random circuit sampling
Yasuhiro Kondo ∗ Ryuhei Mori ∗ Ramis Movassagh † February 4, 2021
Abstract
We prove under the complexity theoretical assumption of the non-collapse of the poly-nomial hierarchy that estimating the output probabilities of random quantum circuitsto within exp( − Ω( m log m )) additive error is hard for any classical computer, where m is the number of gates in the quantum computation. More precisely, we show thatthe above problem is P -hard under BPP NP reduction. In the recent experiments,the quantum circuit has n − qubits and the architecture is a two-dimensional grid ofsize √ n × √ n [AAB + − Ω( n log n ) is hard, and for circuits of depth √ n , for whichthe anti-concentration property holds, estimating the output probabilities to within2 − Ω( n / log n ) is hard. We prove these results from first principles and do not use thestandard techniques such as the Berlekamp–Welch algorithm, the usual Paturi’s lemma,and Rakhmanov’s result. Moore’s law for classical (super-)computers is reaching a saturation point because if the com-putation were done with smaller components confined to smaller spaces, then the quantumeffects would become relevant. Consequently alternative models and architectures are beinginvestigated to empower the future of computation. Among the many proposals, quantumcomputing is currently the only model of computation that could potentially exponentiallyoutperform any classical computer. Proving this in affirmative has been a main driving forcein the field of quantum computation.For quantum computers to have the awesome computational power just described theso called Extended Church-Turing Thesis (ECTT) would need to be refuted. ECTT statesthat a probabilistic Turing machine can efficiently simulate any model of computation thatcan be realized in Nature (i.e., a realistic computation). A single computational task thatwould provably refute ECTT would be sufficient. Therefore it is an imperative near-termgoal to show that for a given computational task (whatever it may be) a quantum computer ∗ Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Japan † IBM Quantum, MIT-IBM AI Research lab, Cambridge MA, U.S.A. P -hard in the worst case [TD04, BJS11], but what is needed is averagecase hardness. The hardness of sampling relies on complexity theoretical assumptions thatpre-date quantum computing. The first proposal for demonstrating sampling-based quantumsupremacy was the original BosonSampling paper of Aaronson and Arkhipov [AA11] in whichthey showed that producing samples from a distribution that mimics the distribution of alinear optical system is hard. Later, Bremner et al showed that a class of circuits known asIQP circuits are also classically hard to sample from [BMS16]. The foremost candidate fordemonstrating quantum supremacy has been the so-called Random Circuit Sampling (RCS)problem, which states that for any classical computer it is hard to produce samples from adistribution that is close to the distribution of a local quantum circuit whose local gates arerandomly and independently are drawn uniformly from the space of all possible gates.Demonstration of quantum supremacy is ultimately given by an experiment for whichthere is solid complexity theoretical evidence of hardness of the task at hand. Indeed Googledid an experiment that involved a random circuit with 53 qubits to demonstrate the hard-ness of RCS [AAB + +
20, HZN + anti-concentration property [HM18], then one can use Stock-meyer’s algorithm [Sto85] to prove that it is sufficient to prove that the amplitudes are hardto estimate to within 2 − n / poly( n ) additive error.The first theoretical evidence for the hardness of computing the output probabilities wasgiven by Bouland et al [BFNV19], who showed that the computation of the amplitudes ofa non-unitary approximation of the actual quantum circuit is hard unless the polynomialhierarchy collapses. Later in [Mov20] it was shown that the probability amplitudes of thethe actual (i.e., unitary) random quantum circuit is hard, and that even estimating the2mplitudes to within 2 − m c , where c is a quantified constant and m is the number of gates,remains P -hard unless the polynomial hierarchy collapses. The validity of the hardness withrespect to additive error estimation is referred to as robustness . We were notified of a draftof an independent work in progress that claims to prove a robustness of 2 − Ω( m ǫ ) [BFLL].As far as we can tell their techniques are very different from ours. In this work we improve the robustness substantially more to 2 − Ω( m log m ) , where m is thenumber of gates. Therefore, our result proves that estimating the probability amplitudes towithin 2 − Ω( n log n ) is hard for constant depth circuit on a grid of √ n × √ n qubits. Circuits ofdepth O ( √ n ), for which anti-concentration is known [HM18], have m = O ( n / ) gates. Andour robustness bound for these circuits is 2 − Ω( n / log n ) .In proving this result, we took an entirely a new approach that does not follow thestandard techniques of the past [AA11, BFNV19, Mov18, Mov20]. In fact we do not use thewell-known Berlekamp–Welch algorithm [WB86], which has instabilities in the presence ofuniform noise. Moreover, we do not use the so called Paturi’s lemma [Pat92], which hasbecome standard in the field for bounding the polynomial extrapolation errors (see below).Our proof also does not make use of Rakhmanov’s result [Rak07], which is usually used toextend error bounds on a discrete set of points to a uniform error bound in a region. Wehowever do rely on the Cayley path introduced in [Mov20] and prove two theorems for thehardness of RCS that complement one another by using oracles of different strengths yetrequiring different success probabilities (Theorems 1 and 2). We prove our hardness resultsfrom first principles and hope that the new approach helps to overcome the insurmount-able difficulties that the standard techniques meet. Our main results are the following twotheorems: Theorem 1.
It is C = P -hard under BPP -reduction to estimate | h | C | i | with − / poly( m ) probability on average over choice of C to within additive error − Ω( m log m ) . Theorem 2.
It is C = P -hard under BPP NP -reduction to estimate | h | C | i | with + n ) probability on average over choice of C to within additive error − Ω( m log m ) .In other word, unless the polynomial hierarchy collapses to finite level, the above task isoutside the polynomial hierarchy. We summarize the proof structure that culminate in Theorems 1 and 2 in Figures 1 and2 respectively.
Our results can be applied to BosonSampling. In an upcoming paper we will extend andapply these techniques to the complexity of BosonSampling [KMM21] for which there was arecent experimental breakthrough [ZWD + − n / poly( n ) forrandom circuits that have the anti-concentration property would be sufficient. These would3. Random Circuit Sampling2. Computing | h | C | i | on averagewith prob. 1 − m ) to withinadditive error 2 − Ω( m log m ) ∴ C = P -hard under BPP -reduction3. Computing | h | C | i | in theworst case to within additiveerror 2 − Ω( m log m ) ∴ C = P -hard under P -reduction4.Any C = P problem BPP NP -reduction [Stockmeyer 1985]Available only when the additive error in Box 2 is 2 − n / poly( n ). BPP -reduction: Lemma 6 P -reduction: Lemma 9Theorem 1 Figure 1: The proof structure and reductions for Theorem 11. Random Circuit Sampling Conjecture: P -hard under BPP NP reduction2. Computing | h | C | i | on averagewith prob. 3 / / poly( n ) to withinadditive error 2 − Ω( m log m ) ∴ P -hard under BPP NP reductions3. Computing | h | C | i | in theworst case to within additiveerror 2 − Ω( m log m ) ∴ P -hard under NP -reduction4.Any P problem BPP NP reduction [Stockmeyer 1985]Available only when the additive error in Box 2 is 2 − n / poly( n ). BPP NP -reduction: Lemma 7 NP -reduction: Lemma 9 and P P ⊆ NP C = P Theorem 2 Figure 2: The proof structure and reductions for Theorem 24orrespond to random circuits on a grid of size √ n × √ n and depth √ n [HM18]. The over-arching goal of proving the quantum supremacy conjecture is achieved (i.e., Corollary 1 isproved) if the following conjecture is proved in the affirmative: Conjecture 1.
Estimating | h | C | i | with probability of + n ) over the choice of quan-tum circuits C to within the additive error − n / poly( n ) implies the collapse of the polynomialhierarchy to a finite level. Assuming Conjecture 1 and anti-concentration of output probabilities, we obtain thehardness of RCS via Stockmeyer’s theorem.
Theorem 3 (Stockmeyer [Sto85]) . Given a Boolean function f : { , } n → { , } , let p = Pr x ∈{ , } n [ f ( x ) = 1] = 2 − n X x ∈{ , } n f ( x ) . Then there exists an
FBPP NP f machine that approximates p to within any multiplicativefactor of / poly( n ) . In conclusion, Theorem 3 along with the anti-concentration property of the output prob-abilities [HM18], and Conjecture 1 prove the following major open problem:
Corollary 1.
Classically sampling from any distribution with a total variation distance of / poly( n ) from the output distribution of the random quantum circuit is hard unless thepolynomial hierarchy collapses to finite level. We now define the interpolation between any two gates of the quantum computation (i.e.,any two unitaries) based on the Cayley path, which was first introduced in [Mov20]. Suppose U , U ∈ U ( N ) are unitary matrices and we wish to interpolate between U and U via apath with nice algebraic properties that can be utilized in our reductions below. Let θ ∈ R and f ( θ ) be the Cayley function f ( θ ) = 1 + iθ − iθ (1)where one defines f ( −∞ ) = −
1. The proposed path is U ( θ ) = U f ( θh ) = N X α =1 f ( θh α ) U | ψ α ih ψ α | (2)where h is a hermitian matrix having the spectral decomposition h = P Nα =1 h α | ψ α ih ψ α | , andsatisfying f ( h ) = U † U . U ( θ ) is a unitary matrix as it is a product of two unitary matrices.Note that U (0) = U f (0) = U and U (1) = U U † U = U as desired. We now derive thealgebraic dependence of the entries of U ( θ ) on θ .5sing the definition of the Cayley function and the foregoing equations we write U ( θ ) = 1 q ( θ ) N X α =1 p α ( θ ) ( U | ψ α ih ψ α | ) , (3)where U | ψ α ih ψ α | are matrices, and q ( θ ) and p α ( θ ) are the polynomials of degree N in θ : q ( θ ) = N Y α =1 (1 − iθh α ) and p α ( θ ) = (1 + iθh α ) Y β ∈ [ N ] \ α (1 − iθh β ) . (4)As before, we define the unitary-valued Cayley path as C k ( θ ) = C k f ( θh k ) (5)where f ( θh k ) is a unitary matrix and h k is hermitian h † k = h k . Suppose C k is a fixed gateof a quantum computation and f ( h k ) is a Haar unitary matrix then C k (0) = C k . Moreover,by the translation invariance of the Haar measure C k (1) = C k f ( h k ) is a Haar random gate.Hence we have an interpolation scheme between any fixed gate and a Haar random gate.In the above equation we can make the dependence on k explicit and denote by p α ( θ ) p k,α ( θ ) and q ( θ ) q k ( θ ). We can now express Eq. (5) as C k ( θ ) = q − k ( θ ) N X α =1 p k,α ( θ ) C k | ψ k,α ih ψ k,α | where p k,α ( θ ) = N X α =1 (1 + iθh k,α ) Y β ∈ [ N ] \ α (1 − iθh k,β ) q k ( θ ) = N Y α =1 (1 − iθh k,α ) . It will be useful to make a change of variables to θ = 1 − x such that generic instancescorrespond to x = 0 and P -hard point to x = +1. Each gate now writes C k ( θ = 1 − x ) = N X α =1 i (1 − x ) h α − i (1 − x ) h α C k | ψ k,α ih ψ k,α | . Let us denote the quantum circuit C ( x ) whose local gates are C k ( x ) by C ( x ) = C m ( x ) · · · C ( x ) C ( x )where C k ( x ) = I ⊗ C k ( x ) is a unitary matrix that only acts non-trivially on the qubit that C k ( x ) acts on.The probability amplitude of starting the quantum computation in the state | n i andmeasuring the string | n i is p ( x ) ≡ |h n | C ( x ) | n i| . Note that at x = 1 we recover the6orst case P -hard instance probability amplitude and x = 0 corresponds to the probabilityamplitude of the generic random circuit. The circuit has the algebraic form [Mov20] |h n | C ( x ) | n i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h n | m Y k =1 C k ( x ) | n i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≡ |h n | P ( x ) | n i| | Q ( x ) | , where (6) P ( x ) ≡ N X α ,...,α m =1 m Y k =1 g k,α k ( x ) [( C k | ψ k,α k ih ψ k,α k | ) ⊗ I ˆ k ] (7) | Q ( x ) | ≡ m Y k =1 N Y α k =1 (cid:12)(cid:12)(cid:12)(cid:12) ix h k,α k r k,α k e iu k,αk (cid:12)(cid:12)(cid:12)(cid:12) . (8) g k,α k ( x ) ≡ (cid:20) e iu k,αk − ix h k,α k r k,α k (cid:21) Y β k ∈ [ N ] \ α k (cid:20) e − iu k,βk + ix h k,β k r k,β k (cid:21) , where we let 1 ± ih k,α k = r k,α k e ± iu k,αk with r k,α k and u k,α k defined as r k,α k = q h k,α k and u k,α k = arctan( h k,α k ).The quantity | Q ( x ) | can be pre-computed in time Θ( m ) as it only depends on theeigenvalues of the local terms which are matrices of size at most 4. Since h k,αk r k,αk < | x | ≤ ∆ = O ( m − ), it is easily seen that | Q ( x ) | is very near one: | Q ( x ) | ≤ m Y k =1 N Y α k =1 (cid:12)(cid:12)(cid:12)(cid:12) ix h k,α k r k,α k e iu k,αk (cid:12)(cid:12)(cid:12)(cid:12) ≤ O ( m ∆) . (9) Our goal here is to prove that estimating p ( x ) ≡ |h n | C ( x ) | n i| to within ǫ additive error is hard for as large an ǫ as possible. That is given an x i and a classical algorithm that promisesto give us p ( x i ) + ǫ i efficiently (polynomial classical time), where | ǫ i | ≤ ǫ ≪
1, we wish toconstruct a low degree algebraic function ˜ p ( x ) whose interpolation to x = 1 is guaranteed tobe hard .Since for any x , | Q ( x ) | can be computed in time Θ( m ) we can reduce the rationalfunctional form of p ( x ) to a polynomial by multiplying through by | Q ( x ) | , which for anygiven x can be treated as simply a constant. Let us denote by the “exact” polynomial p e ( x ) = |h n | P ( x ) | n i| = | Q ( x ) | p ( x ) degree : 8 m, where we treated Q ( x ) as a known constant. Therefore, we have at our disposal a set oftuples ( x i , p e ( x i ) + ǫ i | Q ( x i ) | ). In Eq. (9) we showed that | Q ( x ) | ≤ O ( m ∆), and bytaking ∆ = O ( m − ) we are guaranteed to have | Q ( x ) | ≈
1. This shows that the additiveerror | Q ( x i ) | ǫ i ≈ ǫ i .Let the difference of the exact polynomial p e ( x ) from the one that results from theextrapolation of the erroneous polynomial ˜ p ( x ) be defined by p ( x ) ≡ ˜ p ( x ) − p e ( x ) .
7e are promised that for all | x i | ≤ ∆, | p ( x i ) | ≤ ǫ and wish to show that | p (1) | is sufficientlysmall such it falls within a region whose hardness is guaranteed. We will return to thequantification of this region in Section 4. For now let us bound the polynomial extrapolationerror | p (1) | .Traditionally this bound is obtained using Paturi’s lemma [Pat92], which we recall: Lemma 1. (Paturi’s lemma [Pat92])) Let p ( x ) be a polynomial of degree d , and suppose | p ( x ) | ≤ ǫ for | x | ≤ ∆ where ∆ ∈ (0 , . Then p (1) ≤ ǫ exp[2 d (1 + ∆ − )]For k ≥
0, let us denote by T k ( x ) the k th Chebyshev polynomial, which is a degree k algebraic polynomial defined by T k ( x ) = 12 h ( x + √ x − k + ( x − √ x − k i . Paturi has another result in the same paper (Corollary 2 in [Pat92]), which says
Corollary 2. (Paturi’s Corollary [Pat92])) Let p ( x ) be a polynomial of degree at most d .Assume | p ( x ) | ≤ ǫ in the interval [ − ∆ , ∆] for some < ∆ ≤ . We then have | p ( x ) | ≤ ǫ (cid:12)(cid:12)(cid:12) T d (1 + | x |− ∆∆ ) (cid:12)(cid:12)(cid:12) for all | x | > a . We shall use the latter and prove the following lemma
Lemma 2.
Let p ( x ) be a polynomial of degree d , and suppose | p ( x ) | ≤ ǫ for | x | ≤ ∆ where ∆ ∈ (0 , . Then | p (1) | < ǫ exp( d log | − | ) . Proof.
From Paturi’s corollary we have | p (1) | ≤ ǫ | T d (∆ − ) | . Moreover, | T d ( x ) | ≤ h(cid:12)(cid:12)(cid:12) ( x + √ x − d (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( x − √ x − d (cid:12)(cid:12)(cid:12)i < (cid:16) | x | + √ x − (cid:17) d < | x | d = e d log(2 | x | ) . We conclude that | p (1) | < ǫ exp ( d log(2∆ − )) = ǫ exp h − ( d log ∆) (cid:16) − ∆ (cid:17)i .An issue one faces in making robustness claims is that the discrete bound of | p ( x i ) | ≤ ǫ for | x i | ≤ ǫ , does not readily imply a uniform bound | p ( x ) | ≤ ǫ for all | x | ≤ ∆. This istraditionally remedied by Rakhmanov’s result [Rak07] (see [Mov20, BFNV19]). Here we willdo without Rakhmanov’s result. To do so, take d + 1 points in the interval [ − ∆ , ∆] andestimate the function p ( x ) using the Lagrange interpolation technique. We prove Lemma 3.
Let p ( x ) be a polynomial of degree at most d . Let ∆ ∈ (0 , . Assume that | p ( x j ) | < ǫ for all of the d + 1 equally-spaced points x j = − ∆ + jd ∆ for j = 0 , , , . . . , d .Then | p (1) | < ǫ exp [ d (1 + log ∆ − )] √ πd . (10)8 roof. Let p j = p ( x j ) for all j = { , , , . . . , d } , where by assumption | p j | < ǫ . The Lagrangerepresentation of the function p ( x ) writes p ( x ) = d X j =0 p j δ j ( x ) , δ j ( x ) ≡ Q ℓ = j x − x ℓ Q ℓ = j x j − x ℓ . By triangular inequality we have | p (1) | ≤ ǫ P dj =0 | δ j (1) | . Moreover, using x j = − x d − j andthe fact that | x j | < j we have | δ j (1) | = Q ℓ = j | − x ℓ | Q ℓ = j | x j − x ℓ | = (1 + x j ) Q (1 − x ℓ ) Q ℓ = j | x j − x ℓ | < (1 + x j ) Q ℓ = j | x j − x ℓ | . Since x j − x ℓ = d ( j − ℓ ), we have Q ℓ = j | x j − x ℓ | = ( d ) d Q ℓ = j | j − ℓ | . Moreover Q ℓ = j | j − ℓ | = Q ℓ ∈{ , , ,...,,j − ,j +1 ,...,d } | j − ℓ | = j !( d − j )! and we obtain | δ j (1) | = (cid:18) d (cid:19) d (1 + x j ) Q ℓ = j | j − ℓ | = (cid:18) d (cid:19) d (1 + x j ) j ! ( d − j )! . We express the bound | p (1) | ≤ ǫ P dj =0 | δ j (1) | as | p (1) | < ǫ (cid:18) d (cid:19) d d X j =0 (1 + x j ) j ! ( d − j )! = ǫ (cid:18) d (cid:19) d d ! d X j =0 (cid:18) dj (cid:19) (1 + x j ) . Using the symmetry of x j = − x d − j we have (1 + x j ) + (1 + x d − j ) = 2 and it is easy to seethat (irrespective of the parity of d ) d X j =0 (cid:18) dj (cid:19) (1 + x j ) = 2 d . By Stirling’s inequality n ! ≥ √ πn n n e n , we conclude that | p (1) | < ǫ (cid:18) d (cid:19) d d d ! ≤ ǫ ( e ∆ − ) d √ πd = ǫ exp [ d (1 + log ∆ − )] √ πd . Similarly, we obtain the following lemma where d + 1 points are chosen from L points in[ − ∆ , ∆]. Lemma 4.
Let p ( x ) be a polynomial of degree at most d . Let L be an integer at least d + 1 .Let a , a , . . . , a d be integers satisfying ≤ a < a < · · · < a d ≤ L − . Let ∆ ∈ (0 , .Assume that | p ( x j ) | < ǫ for all of the d + 1 points x j = − ∆ + a j L − ∆ for j = 0 , , , . . . , d .Then | p (1) | ≤ ǫ exp[ d (1 + log((1 + ∆ − ) L − d ))] √ πd . Definition 1.
Let H A be the distribution over circuits with architecture A whose local gatesare drawn independently and at random from the Haar distribution. Let us denote by H A , ∆ the distribution over circuits whose local gates are drawn from distribution induced by theCayley path for θ = 1 − x with | x | ≤ ∆ . Lemma 5. (Total Variation Distance [Mov20]) For a circuit with m gates and an architec-ture A , the total variation distance between H A and H A , ∆ for ∆ = O (1 /m ) is O ( m ∆)In the following, we show the reductions from the worst-case to the average-case com-putation of the output probability of the quantum circuit. While real numbers appear inthe reduction algorithms, they should be represented by poly( m ) bits. The rounding onlycauses additional errors of size 2 − poly( m ) , which will not affect our bounds and conclusions.For simplicity, we ignore rounding issues in the proofs, and work with real numbers. Lemma 6. (Strong oracle) Let O be an oracle satisfying Pr C ∼H A (cid:2) | O ( C ) − |h n | C | n i| | ≤ ǫ (cid:3) ≥ − δ m + 1 then there exists a probabilistic polynomial-time algorithm R with an access to O that forany quantum circuit C satisfies Pr R (cid:2) | R O ( C ) − |h n | C | n i| | < ǫ exp [ O ( m log m )] (cid:3) ≥ − δ − m ) . Proof.
Previously we proved that the total variation distance between H A and H A , ∆ is O ( m ∆) for ∆ = O ( m − ). Hence invoking the oracle O it holds thatPr C ∼H A , ∆ (cid:2) | O ( C ) − |h n | C | n i| | > ǫ (cid:3) = δ m + 1 + O ( m ∆) . By the union bound, the probability that at least one of the 8 m + 1 points evaluated by O has error larger than ǫ is δ + (8 m + 1) O ( m ∆). By choosing ∆ = Θ( m − k ) for some constant k >
2, from Lemma 5, the error probability is at most δ + 1 / poly( m ). From Lemma 3 weknow that if all 8 m + 1 evaluation points have an error at most ǫ , then the extrapolationerror from x ∈ [ − ∆ , ∆] to x = 1 in the Lagrange extrapolation is given (via Eqs. (9) and(10)) ǫ | Q (1 + ∆) | exp [8 m (1 + log ∆ − )] √ πm ≤ ǫ (1 + O ( m ∆)) exp [8 m (1 + log ∆ − )] √ πm . Since ∆ = Θ( m − k ) for some constant k >
2, we obtain the Lemma.It is desirable to make the oracle O as weak as possible. In the following Lemma we showthat this can be done at the expense of introducing an NP -machine.10 lgorithm 1 The reduction algorithm R where d := 8 m and L := ⌈ ( d + 1) /δ ⌉ function R ( C )Draw H according to H A for i ∈ { , . . . , L − } do x i ← ∆(2 i − L + 1) / ( L − y i ← O ( C H ( x i )) | Q ( x i ) | l ← r ← loop poly( m ) times c ← ( l + r ) / if W (cid:0) d , ( x i , y i ) L − i =0 , l, c (cid:1) then r ← c else l ← c return l Lemma 7. (Weak oracle) Let O be an oracle satisfying Pr C ∼H A (cid:2) | O ( C ) − |h n | C | n i| | ≤ ǫ (cid:3) ≥
34 + δ. Then there exists a probabilistic algorithm R with running time poly( n, m ) accessing O andan NP -machine that for any quantum circuit C satisfies Pr R (cid:2) | R O , NP ( C ) − |h n | C | n i| | < ǫ exp [ O ( m log m )] (cid:3) ≥
12 + δ − m ) . Proof.
We first describe the probabilistic algorithm R for the reduction. Then, we show thatthe algorithm R satisfies the conditions in the lemma. First, define the NP problem W asfollows. (Input) A positive integer d in the unary representation, L pairs { ( x i , y i ) ∈ R } i ∈{ , ,...,L − } .Two rational numbers, l, r ∈ R . (Output) True, if there exists a polynomial ˜ p ( x ) = P dj =0 a j x j such that a j ∈ R for all j = 0 , . . . , d , (cid:12)(cid:12) { i ∈ { , . . . , L − } | | ˜ p ( x i ) − y i | ≤ | Q ( x i ) | ǫ } (cid:12)(cid:12) ≥ (1 + δ ) L/ , and ˜ p (1) = a ∈ [ l, r ). False, otherwise.Obviously, this problem is in NP since for a given certificate ˜ p ( x ), the conditions above canbe verified in polynomial time. We now describe the reduction algorithm R in Algorithm 1.Then, we will show that the algorithm R satisfies the conditions in the lemma.Obviously, R is a probabilistic polynomial-time algorithm accessing the oracle O andthe NP -oracle W . We set ∆ = Θ( m − k ) for some constant k > H A and H A , ∆ is O ( m ∆) = 1 / poly( m ) from Lemma 5. Using Markov’sinequality the probability that at least (1 + δ ) L/ ǫ is at least 1 / δ − / poly( m ) (see [Mov20] for details). Assume that at least (1 + δ ) L/ ǫ . In other words, (cid:12)(cid:12) { i ∈ { , . . . , L − } | | p ( x i ) − y i | ≤ | Q ( x i ) | ǫ } (cid:12)(cid:12) ≥ (1 + δ ) L/ d polynomial p ( x ) := | h n | P ( x ) | n i | . In this case, W (cid:0) d , ( x i , y i ) L − i =0 , , (cid:1) )is true since p ( x ) satisfies the conditions and | Q ( x i ) | ≤
2. Then, there exists a polynomial e p ( x ) = P dj =0 a j x j such that a j ∈ R for all j = 0 , . . . , d , (cid:12)(cid:12) { i ∈ { , . . . , L − } | | ˜ p ( x i ) − y i | ≤ | Q ( x i ) | ǫ } (cid:12)(cid:12) ≥ (1 + δ ) L/ , and | ˜ p (1) − R ( C ) | ≤ − poly( m ) .In the following, we show that | ˜ p (1) − p (1) | ≤ ǫ (1 + O ( m ∆)) exp { d (1 + log ∆ − ) } . Definethe two sets S p and S ˜ p by S p = (cid:8) i ∈ { , . . . , L − } | | p ( x i ) − y i | ≤ | Q ( x i ) | ǫ (cid:9) ,S ˜ p = (cid:8) i ∈ { , . . . , L − } | | ˜ p ( x i ) − y i | ≤ | Q ( x i ) | ǫ (cid:9) . Since S p and S ˜ p have sizes that are at least (1+ δ ) L/
2, then | S p ∩ S ˜ p | = | S p | + | S ˜ p |−| S p ∪ S ˜ p | ≥ (1 + δ ) L − L ≥ δL ≥ d + 1.Since p ( x ) and ˜ p ( x ) are degree d polynomials by assumption, and | p ( x i ) − ˜ p ( x i ) | ≤| p ( x i ) − y i | + | y i − ˜ p ( x i ) | ≤ | Q ( x i ) | ǫ for at least d + 1 points x , . . . , x L − ∈ [ − ∆ , ∆], weobtain the desired result | p (1) − ˜ p (1) | ≤ ǫ (2 + O ( m ∆)) exp (cid:20) d (cid:18) (cid:18) (1 + ∆ − ) L − d (cid:19)(cid:19)(cid:21) from Lemma 4. Hence, | p (1) − R (1) | ≤ ǫ (2 + O ( m ∆)) exp [ d (1 + log((1 + ∆ − )( L − /d ))] +2 − poly( m ) . Since ∆ = Θ( m − k ) for some constant k >
1, we obtain the Lemma.If we take δ = 1 / poly( n ), the success probability of 1 / δ can be boosted to a constantgreater than 1 / O ( δ − ) calls to the oracle. C = P -hardness of the computation of probability am-plitude for the worst-case quantum circuit Following [FGHP98], we define the class
GapP of functions.
Definition 2 ( GapP function) . Given a language L ⊆ Σ ∗ , let L x = { y ∈ Σ ∗ | h x, y i ∈ L } . Afunction f : { , } ∗ → Z is in GapP if there is a language L in P and an integer k such that f ( x ) = | Σ n k ∩ L x | − | Σ n k − L x | where n = | x | . GapP function basically is the difference of accepting and rejecting paths. Following[FGHP98] we define the following languages
Definition 3 ( C = P ) . A language L is in the class C = P if there is a GapP function f suchthat for any x , x ∈ L if and only if f ( x ) = 0 . Fenner et al proved the following lemma.
Lemma 8. (Equivalent to Thm 3.2 in Fenner et al [FGHP98]) For any f ∈ GapP there is apoly-time uniform family of quantum circuits { C n ( x ) } and a polynomial p such that for all x of length n , |h n | C n ( x ) | n i| = f ( x )2 p ( n ) . Proof.
For any f ∈ GapP , there is a function F ( x, y ) such that F ( x, y ) is computable inpolynomial time and f ( x ) = ( |{ y ∈ { , } m | F ( x, y ) = 0 }|−|{ y ∈ { , } m | F ( x, y ) = 1 }| ) / C n ( x ) = H ⊗ m V x H ⊗ m where V x = X y ∈{ , } m ( − F ( x,y ) | y ih y | . Then h n | C n ( x ) | n i = 12 m X y ∈{ , } m ( − F ( x,y ) = 12 m ( |{ y | F ( x, y ) = 0 }| − |{ y | F ( x, y ) = 1 }| ) ≡ f ( x )2 m − , It follows that |h n | C n ( x ) | n i| = f ( x )2 m − and p ( n ) = 2 m − P O , where O is an oracle that estimates the probability amplitudesto small additive errors, can solve C = P problems. Lemma 9.
If there exists an oracle O that, for any given quantum circuit C , can compute |h n | C | n i| to within additive error − n ǫ for some constant ǫ > . Then there is a P O algorithm that solves any C = P problem.Proof. Since f ( x ) ∈ N for any problem in C = P , the key is to distinguish between f ( x ) = 0and f ( x ) ≥
1. For any ǫ > (cid:12)(cid:12) h m + k | C n ( x ) ⊗ I ⊗ k | m + k i (cid:12)(cid:12) = |h m | C n ( x ) | m i| for any positive integer k , we can call O with the enlarged quantum circuit C n ( x ) ⊗ I ⊗ k toapproximate |h m | C n ( x ) | m i| to within additive error 2 − ( m + k ) ǫ . From the previous result(Lemma 8) we have that |h m | C n ( x ) | m i| = 2 − m +2 f ( x ). To distinguish f ( x ) = 0 from f ( x ) = 0 it is sufficient to set 2 − ( m + k ) ǫ ≤ − m . So we take k = (2 m ) /ǫ − m . This meansthat there is a P O algorithm that distinguishes f ( x ) = 0 from f ( x ) = 0, hence there is a P O algorithm that solves any C = P problem.These prove our main theorems which we restated along with their proofs:13 heorem 1. It is C = P -hard under BPP -reduction to estimate | h | C | i | with − O (1 /m ) probability on average over choice of C to within additive error − Ω( m log m ) .Proof. This is immediate from Lemmas 6 and 9.
Theorem 2.
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AppendixA Proof of Lemma 4
Let p j = p ( x j ) for all j = { , , , . . . , d } , where by assumption | p j | < ǫ . The Lagrangerepresentation of the function p ( x ) writes p ( x ) = d X j =0 p j δ j ( x ) , δ j ( x ) ≡ Q ℓ = j x − x ℓ Q ℓ = j x j − x ℓ . By triangular inequality we have | p (1) | ≤ ǫ P dj =0 | δ j (1) | . Moreover, using the fact that | x j | ≤ ∆ for all j we have | δ j (1) | = Q ℓ = j | − x ℓ | Q ℓ = j | x j − x ℓ | < Q ℓ = j (1 + ∆) Q ℓ = j | x j − x ℓ | < (1 + ∆) d Q ℓ = j | x j − x ℓ | . Since | x j − x ℓ | = L − | a j − a ℓ | ≥ L − | j − ℓ | , we have Q ℓ = j | x j − x ℓ | ≥ ( L − ) d Q ℓ = j | j − ℓ | .Moreover Q ℓ = j | j − ℓ | = Q ℓ ∈{ , , ,...,,j − ,j +1 ,...,d } | j − ℓ | = j !( d − j )! and we obtain | δ j (1) | < (cid:18) L − (cid:19) d (1 + ∆) d Q ℓ = j | j − ℓ | = (cid:18) L − (cid:19) d (1 + ∆) d j ! ( d − j )! . We express the bound | p (1) | ≤ ǫ P dj =0 | δ j (1) | as | p (1) | < ǫ (cid:18) ( L − (cid:19) d d X j =0 j ! ( d − j )! = ǫ (cid:18) ( L − (cid:19) d d ! d X j =0 (cid:18) dj (cid:19) = ǫ (cid:18) ( L − (cid:19) d d d ! . By Stirling’s inequality n ! ≥ √ πn n n e n , we conclude that | p (1) | < ǫ (cid:18) ( L − (cid:19) d d d ! ≤ ǫ ( e ( L − − ) /d ) d √ πd = ǫ exp[ d (1 + log((1 + ∆ − ) L − d ))] √ πd .πd .