Sufficient condition on noise correltions for sclble quntum computing John Presill, 2 Februry 202 Is quntum computing sclble? The ccurcy threshold theorem for quntum computtion estblishes tht sclbility is chievble if the noise fflicting the quntum computer is not too strong nd not too strongly correlted. For sclbility to fil s mtter of principle then, either the ccepted principles of quntum physics must fil for complex highly entngled systems (s t Hooft [th99] hs suggested, or else either the Hmiltonin or the quntum stte of the world must impose noise correltions tht overwhelm fult-tolernt quntum protocols (s Kli [K] hs suggested. To get further insight into this issue, it is useful to derive conditions on the Hmiltonin tht suffice for sclbility. The noise model we consider is formulted by specifying time-dependent Hmiltonin H tht governs the joint evolution of the system nd the bth, which cn be expressed s H = H S + H B + H SB ; ( here H S is the time-dependent Hmiltonin of the system tht relizes n idel quntum circuit, H B is the Hmiltonin of the bth, nd H SB, which describes the coupling of the system to the bth, is the origin of the noise. We plce no restrictions on the bth Hmiltonin H B. Without ny loss of generlity, we my expnd the system-bth Hmiltonin in the form H SB = = i i,i 2,...i H ( i i,i 2,...i = =! i i,i 2,...i H ( i i,i 2,...i. (2 Here, H ( i i,i 2,...i cts on the qubits lbeled by the indices i i, i 2,...i, nd lso cts on the bth; for ech we sum over ll wys of choosing system qubits. We use i i, i 2,...i to denote n unordered set of qubits; by definition, H ( i i,i 2,...i is invrint under permuttions of the qubits nd vnishes if two of the indices coincide. Hence the two expressions for H SB re equivlent. We will not need to ssume nything bout the initil stte of the bth, except tht the system qubits cn be well enough isolted from the bth tht we cn prepre single-qubit sttes with resonble fidelity. We use the term loction to spe of n opertion in quntum circuit performed in single time step; loction my be single-qubit or multi-qubit gte, qubit preprtion step, qubit mesurement, or the identity opertion in the cse qubit is idle during time step. We model noisy preprtion s n idel preprtion followed by evolution governed by H, nd noisy mesurement s n idel mesurement preceded by evolution governed by H. It is convenient to imgine tht ll system qubits re prepred t the very beginning of the computtion nd mesured t the very end; in tht cse the noisy computtion cn be fully chrcterized by unitry evolution opertor U cting jointly on the system nd the bth, obtined by solving the time-dependent Schrödinger eqution for the Hmiltonin Eq. (2. Our gol is to derive from Eq. (2 n expression for the effective noise strength ε of the noisy computtion, which is defined s follows. We envision performing forml expnsion of U in powers of the perturbtion H SB, to ll orders. Consider prticulr set I r of r circuit loctions, nd let E(I r denote the sum of ll terms in the expnsion such tht every loction in I r is fulty, i.e., such tht t lest one of the qubits t tht loction is struc t lest once by term in H SB during the execution of the gte. We sy tht the noise hs effective noise strength ε if E(I r ε r (3 for ny set I r. The ccurcy threshold theorem for quntum computing shows tht sclble quntum computing is possible if ε is less thn positive constnt ε 0 [AGP2006, NP2009]. Let us define η ( = mx H ( i i,i 2,i 3,...,i t 0, (4 i 2,i 3,...,i
the mximum is over ll qubits nd ll times, nd t 0 is the mximl durtion of ny loction. Then our min result n be stted s follows. Theorem. (Effective noise strength for correlted Hmiltonin noise If ech quntum gte cts on t most m qubits nd if η ( f α, (5 for ll, then ε 2mα exp g = f + l= ( = g, (6 (!f +l (2α l. (7 ( + l! It follows tht quntum computing is sclble provided the strength of -qubit interctions decys sufficiently rpidly with (so tht the sums in Eq. (6 nd Eq. (7 converge, nd lso decys s the sptil seprtion of the qubits increses (so tht the sum defining η ( in Eq. (4 converges. If, for exmple, f =, then nd hence g (2α l = ( 2α C(α, (8 ( ε 2mα e (e /2 C(α 4.72 mα, (9 the lst pproximtion uses C(α for α, s is the cse if ε is smller thn the threshold vlue ε 0 0 4. If insted f!/ p, (0 then ( g! ( (!f +l (2α l p =! (!( + l! p (2α l ( + l!f p ( + l!!( + l p (! ( ( + l(2α l =! ( + l (2α l p p ( ( =! p 2α + 2α ( 2α 2! ( 2 = C(α 2! p 2α p. ( we obtined the inequlity in the lst line using. For p = 2, for exmple, we find ( C(α 2 ε 2mα e π2 /2 4.55 mα, (2 gin using C(α to obtin the numericl expression. For p > the sum over in Eq. (6 converges, nd hence we obtin finite expression for ε. Therefore, sclble fult-tolernt quntum computtion is chievble for sufficiently smll (positive vlue of α. 2
In [AKP06], sclbility ws proven for the specil cse in which only the = 2 term in the Hmiltonin is nonzero. To prove Theorem we generlize the ides used in [AKP06]. We write the system-bth Hmiltonin s H SB = H SB, (3 is shorthnd for the indices, nd i i, i 2,...i in Eq. (2. For the se of conceptul clrity we imgine dividing time into infinitesiml intervls, ech of width, nd express the time evolution opertor for the intervl (t, t + s U(t +, t e i H e i HS e i HB (I SB i H SB,. (4 (We hve omitted terms higher order in H ; strictly speing, then, to justify Eq. (4 we should regulte the bth Hmiltonin by imposing n upper bound on its norm, then choose smll enough so these higher order terms cn be sfely neglected. We expnd U(t +, t s sum of monomils, for ech vlue of either I SB or i H SB, ppers; then we obtin the perturbtion expnsion of the full time evolution opertor U over time T by stitching together T/ such infinitesiml time evolution opertors. We will refer to the r specified loctions in the set I r s the mred loctions nd to the remining loctions s the unmred loctions. For now, suppose for definiteness tht ll of the mred loctions re single-qubit gtes. For ny term in the perturbtion expnsion contributing to E(I r there must be n erliest infinitesiml time intervl in ech of the r mred loctions term H SB, cts nontrivilly on tht qubit. Suppose we fix the infinitesiml time intervls these erliest insertions of H SB occur, nd lso fix the terms {H SB, } in the system-bth Hmiltonin tht ct there, but sum over ll the terms in the perturbtion expnsion cting in other time intervls nd on other qubits. Then in between the fixed erliest insertions in the mred loctions, the joint evolution of the system nd the bth is governed by modified Hmiltonin (modified H (modified = H S + H B + H (modified SB, H (modified SB = H SB,, (5 the modified sum excludes ny term H SB, cting nontrivilly on ny one of the mred loctions during ny time intervl prior to the fixed time of the erliest insertion. The importnt point is tht the time evolution opertor in between successive insertions of the perturbtion is unitry nd hence hs unit opertor norm. We conclude, then, tht the contribution to E(I r with the erliest insertions t the mred loctions fixed hs opertor norm bounded bove by (erliest ( H SB,, (6 the product is over the terms in the system-bth Hmiltonin tht ct t the erliest insertions. To bound E(I r, we sum over the t 0 / time intervls t ech loction the erliest insertion my occur, nd lso sum over ll the wys of choosing the term H SB, tht cts t ech insertion, obtining E(I r (insertions {H SB,} (erliest ( H SB, t 0. (7 Summing over the possible intervls for the first insertion turns the fctor into the fctor t 0. Now we hve to figure out how to sum over ll wys of choosing the terms {H SB, } cting t the erliest insertions inside the r mred circuit loctions. Since H SB contins multi-qubit terms, single term in H SB cn simultneously produce the first insertion t multiple circuit loctions occurring in the sme time step. 3
Specificlly, single term in H ( might cuse simultneous fults in j of the r mred loctions for ny j. We use the term j-contrction to refer to the cse when single term in Eq. (2 produces the first insertion in ech of j mred loctions. The strength of one-contrction cn be bounded by η (+l = mx H (+l i j,j 2,...,j l η = i,j,j 2,...,j l t 0 = mx l! η (+l, (8 i j,j 2,...,j l H (+l i,j,j 2,...,j l t 0 (9 Here i is one of the mred loctions, nd other indices re summed over ll loctions (both mred nd unmred, to llow for the possibility tht the insertion t mred loction need not be the first insertion; hence the higher order (l > 0 terms in Eq. (9 cn be contributions to the strength of the one-contrction rther thn j-contrction for j > even though some of the loctions in {j, j 2,..., j l } my be mred. Similrly, for >, the strength of -contrction cn be bounded by = η = i,i 2,...,i j,j 2,...,j l, (20 H (+l i,i 2,...i,j,j 2,...,j l t 0. (2 Here for the in sum the qubits re restricted to the mred loctions nd for the ll sum they my be t either mred or unmred loctions. By summing ll wys of choosing the first insertion in ech of r mred loctions, we obtin the bound ε r (r r,r 2,r 3,... = r! (η r. (22 Here r is the number of -contrctions, nd the sum (r is subject to the constrint r = r. To obtin Eq. (22, we observe tht H ( i,i 2,...i i,i 2,...,i r (23 contins ech wy of choosing r -contrctions mong the r mred loctions r! times, plus dditionl nonnegtive terms; the fctor /r! in Eq. (22 compenstes for this overcounting. To go further we wish to relte η for > to η. Note tht in for >, we cn replce the sum over wys to choose qubits by sum over ll qubits divided by!, nd similrly we cn replce the sum over the wys to choose l qubits by sum over ll qubits divided by l!, obtining =! i,i 2,...,i l! j,j 2,...,j l H (+l i,i 2,...i,j,j 2,...,j l t 0. (24 4
By summing i i over the r mred loctions we obtin the bound r mx i! i 2,...,i l! j,j 2,...,j l H (+l i,i 2,...i,j,j 2,...,j l t 0 ; (25 note tht we still hve bound if we extend the in sum to sum over ll qubits. From Eq. (9 we hve = mx i ( + l! i 2,...,i which implies (for > ( + l! r = r ( + l! l! l Hence we find (for > η r 2 Now suppose, s in the hypothesis of Theorem, tht From Eq. (28 we obtin η ( j,j 2,...,j l H (+l i,i 2,...i,j,j 2,...,j l t 0. (26 r 2+l. (27 2 +l. (28 (! η( f α (!. (29 g = f + η rg (2α, (30 l= (!f +l (2α l. (3 ( + l! Then the bound Eq. (22 becomes ε r (r r,r 2,r 3,... = r! ( rg (2α r = (2α r (r r,r 2,r 3,... = reclling the constrint on the sum. If we now relx the constrint on the sum, we hve = r! ( rg r, (32 ε r (2α r ( rg r ( r rg = (2α exp r = r! =0 = ( r ( ( r = (2α (exp r g g = 2α exp. (33 Finlly, we observe tht in the cse of n m-qubit gte loction, the loction is fulty if the system-bth perturbtion cts nontrivilly on ny one of m qubits, which enhnces the noise strength by the fctor m. This completes the proof of Theorem. = 5
If the hypotheses of Theorem fil, we need not despir. In [NP09] it is shown tht sclbility my still be provble if we me further ssumptions bout the stte of the bth (which ws ssumed there to be Gussin stte in which sptil correltions decy sufficiently rpidly. We might insted suppress the noise correltions using specilized methods tht re not incorported into the stndrd proofs of the quntum threshold theorem, such s dynmicl decoupling. Chrcterizing the residul noise correltions when dynmicl decoupling is employed seems to be hrd problem, though some preliminry steps were ten in [NLP]. References [AKP06] D. Ahronov, A. Kitev, nd J. Presill, Fult-tolernt quntum computtion with long-rnge correlted noise, Phys. Rev. Lett. 96, 050504 (2006. [AGP06] P. Aliferis, D. Gottesmn, nd J. Presill, Quntum ccurcy threshold for conctented distnce-3 codes, Quntum Informtion nd Computtion 6, 097-65 (2006. [th99] G t Hooft, Quntum grvity s dissiptive deterministic system, Clssicl nd Quntum Grvity 6, 3263-3279 (999. [K] G. Kli, How quntum computers fil: quntum codes, correltions in physicl systems, nd noise ccumultion, rxiv:06.0485 (20. [NLP] H.-K. Ng, D. A. Lidr, nd J. Presill, Combining dynmicl decoupling nd fult-tolernt quntum computtion, Phys. Rev. A 84, 02305 (20. [NP09] H.-K. Ng nd J. Presill, Fult-tolernt quntum computtion versus Gussin noise, Phys. Rev. A 79, 03238 (2009. 6