* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Poster PDF (3.9mb)
Bell test experiments wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Quantum field theory wikipedia , lookup
Particle in a box wikipedia , lookup
Quantum decoherence wikipedia , lookup
Density matrix wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Quantum dot wikipedia , lookup
Scalar field theory wikipedia , lookup
Algorithmic cooling wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum entanglement wikipedia , lookup
Coherent states wikipedia , lookup
Quantum fiction wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Bell's theorem wikipedia , lookup
Boson sampling wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Ultrafast laser spectroscopy wikipedia , lookup
Probability amplitude wikipedia , lookup
EPR paradox wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
History of quantum field theory wikipedia , lookup
Path integral formulation wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum group wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Quantum state wikipedia , lookup
Quantum key distribution wikipedia , lookup
Hidden variable theory wikipedia , lookup
Quantum teleportation wikipedia , lookup
Quantum machine learning wikipedia , lookup
Algorithms for a Scalable Quantum Computer Optimal Pulse Sequences, Montgomery Factoring, and Bayesian Inference Guang Hao Low, Rich Rines, Theodore Yoder, Isaac Chuang Montgomery Factoring Introduction Bayesian Inference Quantum computers promise speedups to a whole host of classical algorithms, from exponential advantage in factoring to a plethora of polynomial advantages in machine learning tasks. However, as the size of the problem grows, the quantum circuit required to solve it also grows, and small errors in single gates add to have a substantial effect on the output. Scaling up a quantum computer therefore requires either simplifying the circuits or reducing errors. To that end, we present open-loop error correction for single-qubit gates in the form of pulse sequences of optimal length. We also show how to reduce the circuit for Shor’s algorithm using the Montgomery product from number theory, bringing the factoring of 35 within experimental grasp. Finally, we demonstrate how a Bayesian inference task can be performed with polynomial advantage on a scalable quantum computer. Say that we have a probability distribution P (~ x) on n binary variables. After observing some set of evidence variables E = e how can we sample from P (Q|e) for a set of query variables? Such sampling allows us to perform approximate inference, inferring the most likely values of Q given e by evaluating argmaxQ P (Q|e). X1 X2 X3 P (x2 |x1 ) P (x3 |x1 ) Say that we wish to perform a single-qubit rotation R [✓] by = X̂ cos + Ŷ sin but we only angle ✓ around the axis have access to a faulty operation M [✓] = R [✓(1 + ✏)] which over- or under-rotates by a factor ✏ . In the presence of these amplitude errors, how can we still perform R [✓] accurately? The answer comes from the non-commutativity of single-qubit operations which allows us to chain faulty pulses in sequence such that L Y n+1 U0 [✓] = M'j [✓0 ] ·M0 [✓] = R0 [✓] + O(✏ ). (1) j=1 | {z } ⌘S(~ ') ? E =e N ⇥ O(nm) q-sample state [3]: | O(N n) Pi q-sample | Pi ) N = O(1/P (e)) X p = x1 ,...,xn Figure 3: (Right) This circuit prepares the qsample | P i based on the structure of the x1 Bayes net in Fig. 2. Once compiled, the circuit complexity is O(n2m ). (Below) The quantum x2 rejection sampling algorithm produces one sample of P (Q|e) per q-sample prepared. It does so by enhancing the component of | P i x3 that has the correct evidence |ei using amplitude amplification. This effectively x4 speeds up the classical post-selection by a square-root. Quantum Rejection Sampling 1⇥ (5) represents the computational bottleneck of Shor’s algorithm. Montgomery reduction is a commonly used classical computational technique for efficient modular exponentiation under a large modulus, reducing the algorithmic complexity of modular reduction by division to that of multiplication: c Binary modular addition of a constant for shift-and-add (S+A) multiplication is assumed to have the circuit cost fo constant additions. Remarkably, we multiplication can reclaimcircuit. the algorithmic advantage by adapting d The Zalka/Kutin method is an approximate to a uniquely quantum mechanical arithmetic model acting in various multiplication procedures.multiplier Gate counts and depths are measured in quantumofFourier space. The resulting provides a unique P (x1 , . . . , xn )|x1 , . . . , xn i TABLE (4) I: Circuit characteristics of multi-qubit gates. scalable building block for constructing Shor’s circuits: |0i U1 U2 |0i U4 |0i Q|e i|ei p m O(n2 / P (e)) | U3 |0i Amp. Amp. | 3-Qub 3n 3n O n2 O n2 0 0 ability to implicitly calculate U with no computational overhead. a For the QFMP circuit costs, some deterministic optimizations were made, such as combining sequential controlled rot Unfortunately, this implicit calculation is not immediately reversible, same control and target qubits b For n-bit binary arithmetic mitigating advantage of size Montgomery reductionaddition when and naively circuits, the we assume a circuit of 6n for non-modular 3n for non-modula q ⇠ P (Q|e) O(N P (e))⇥ adapted to quantum modular exponentiation. constant post-select N ⇥ x ⇠ P (x) The dependence of L on n that is implicit in this equation is crucial. We prove that L > n and argue that L = 2n is achievable. This is in contrast to the next best sequences for arbitrary ⌘ ✓/2⇡, which scale as L = O(n3.09 ) [1]. |xi ! |a x mod N i, Figure 2: (Left) A Bayes net is a directed acyclic graph with n nodes and maximum indegree m showing the Given: T = xy < N R conditional dependences of random variables of a 0 1 x) (Eq.(3)). With each node Multiplication N := N mod R Depth Qubits probability distribution P (~ is Method Gate Count Parallelized stored a table of probabilities. Sampling nodes top to 0 (n) QFMP, Simplea 10n2 9n + 2 O 2n U := T N mod R x) in time O(nm). bottom, gives a sample of P (~ 2 QFMP, General 12n + 3n + 6 O1(n) 2n + 2 (Below) Classical rejection sampling takes N samples (T +18n UN (mod N ) 2 )/R ⌘ T R from P (~ x) and rejects those with incorrect evidence Quantum Montgomery, Generalb + O (n) O n2 3n + O (1) E` 6= e . Since the correct evidence only appearsS+A, with 2 )/R < 2N UN X4 Ripple-Carry Addersc [18] (T [12]+24n + O (n) O n2 2n + O (1) probability P (e) , if we desire one sample from the 3 2 2 S+A, QFT Adders [4] [2] 2n + O n O n 2n + 2 P (Q|e) posterior we must set and N = O(1/P (e)) P (x4 |x1 , x3 ) The classical advantage of Montgomery reduction results from the d 2 take O(nm/P (e)) time. Zalka/Kutin [18] [6] 2n + O (n log2 n) O (n) 3n + O (log2 n) Classical Rejection Sampling Optimal Pulse Sequences 2k (3) P (~x) = P (x1 )P (x2 |x1 )P (x3 |x1 )P (x4 |x1 , x3 ) P (x1 ) Repeated application of the controlled modular product operation, 1⇥ q ⇠ P (Q|e) Pi Multiplication Method Fourier Montgomery Shift-and-Add [4] Fourier Shift-and-Add, Draper [5] Fourier Shift-and-Add, Beauregard [6] Depth O (n) O n2 O n2 O n3 Gates 12n2 + 3n + 6 24n2 + O (n) 2n3 + O n2 2n3 + O n2 Qubits 2n + 2 2n + O (1) 3n 2n + 2 Small circuit width and small CONCLUSIONS lower-order terms in the circuit depth VIII. and gate count make the Fourier Montgomery method immediately applicableperforms to foreseeable factoring experiments, such as linear N = 35. The overall QFMP operation Montgomery modular multiplication with circuit depth width of 2n + 2 qubits. For the uncontrolled operator, no three-qubit gates are required; however 3n wi for the controllable version O(n2m ) |p3 ! in the ci |0! required for modular exponentiation. Further, small constant terms depth, a low quadratic gate count, and the ability to perform Montgomeryization within classical pre |p2 ! N . This |0! advantage of the operator to exist both asymptotically and for small allows the computational + N2 QF T † N introduces opportunities |p1 ! |0! for quantum factoring+ circuits. 2 N We notice that, in practice, the QFMP+operation will likely be more efficient than the version presente 2 N + |0! 0 ! approxim as all of our operations are wrapped2 up in Fourier rotations, we note that for larger N we |pcan ×y angles to greatly reduce output [1]. We also notice tha N |a ! |x3 !circuit size with little change •in the− experimental 3 2 N gates required to make the QFMP controllable are acting on a subset − 2 of the possible Hilbert space of thre j N ~=' ~R (' ~= ' ~ R ) result in Im L (~ The symmetries ' ') = 0 for j • • − |x ! 2 ! state. Th instance, one of the three in the|a|0i 2 qubits input to the initial Fredkin gates is deterministically 2 ÷y even (odd) and the resulting sequences are called initialed PD the (AP). opportunity for hardware-specific optimization, including potentially breaking the operation into tw • • |a1 ! |x1 ! All such sequences have L = 2n. These optimizations are very specific to the particular implementation of the QFMP operation, and the • • |a0 ! |x0 ! thoroughly explored here. SolvingMethods for Sequences We introduce three techniques for solving (2): • Analytical: The substitution tk = tan( k /2) converts (2) into a system of polynomial equations, which can be solved by Groebner [1] Adriano Barenco, Artur Ekert, Kalle-Antti Suominen, and Päivi Törmä. Approximate quantum fourier bases to yield AP1, 2, 3 and PD2, 4. @ j Phys. Rev. A, 54:139–146, Jul 1996. • Perturbative: Given a solution at 0 and nonzero Jacobian @'k Ldecoherence. 0 [2] Stephane Beauregard. Circuit for shor’s algorithm using 2n+3 qubits. arXiv, 2003. http://arxiv.o solutions '( ) for ⇡ 0 can be obtained. For a certain AP 2 ph/0205095v3. 1. Ken Brown, Aram Harrow, Isaac Chuang, Phys. Rev. A 70, Figure 1: We compare the transition probability p = |h0|U0 [⇡]|1i| effected by ToP sequence ToP, 'k (0) = ⇡ and we prove a nonzero Jacobian[3] at C. 0. H. Bennett. Logical 052318 reversibility of computation. IBM J. Res. Dev., 17:525 – 532, Nov 1973. pulse sequences with amplitude error ✏ in the cases of length L = 9 (blue) and (2004). [4] Thomas G. Draper. Addition on a quantum computer. arXiv, 2000. http://arxiv.org/abs/quant-ph/00080 also L = 25 (red). Included are BB4,12 of this work, Vitanov’s V8,24 , • Numerical: We use Mathematica [2] to find roots to equation (2) up 2. Wolfram Research Inc., Mathematica, Version 9.0, 2 Shaka’s 2, Cho’s C9 , Husain’s F2 , and Tycko’s S2 . [5] Lieven M. K. Vandersypen et. al. Experimental realization of shors quantum factoring algorithm using nu to n = 12. IL (2012). resonance. arXiv, 2001.Champaign, http://arxiv.org/abs/quant-ph/0112176. 3. Dorit Aharonov, Amnon Ta-Shma, STOC ACM, pp. 20-29 Name Length Notes Our sequences are obtained algebraicallya pulse by expanding We of L faulty pulses M'j [✓j ]. [6] Samuel A. Kutin. Shor’s algorithm on a nearest-neighbor machine. arXiv,’03, 2006. http://arxiv.org/abs/quantsequence S (1). consisting (2003). [7] Igor L. Markov and Mehdi Saeedi. Constant optimized quantum circuits for modular multiplication and e Denote simplify by ' ~ the the vector of phase angles ('1 , '2 , . . . , 'L ), SCROFULOUS 3 n = 1, non-uniform ✓j [9] make the assumption ✓0 = 2⇡ to dramatically result. 2 n which are our free parameters. Leaving each amplitude ✓ arXiv, 2012. http://arxiv.org/abs/1202.6614. 4. Christof Zalka, arXiv:quant-ph/9806084 (1998) j Pn, Bn O(e ) Closed-form [5] ~ are The resulting n complex equations to be solved for ' as a free parameter (e.g. SCROFULOUS [9]) may help [8] Alfred J. Menezes, Scott A. Vanstone, and arXiv:quant-ph/0008033 Paul C. Van Oorschot. Handbook of Applied Cryptography. CR 5. Thomas Draper, (2000) 3 n 30, numerical [5] SKn O(n ) j j reduce sequence length, but we find that a fixed value Boca Raton, FL, USA, 1st edition, 1996. n > 30, conjectured (2) (~ ' ) = f ( ), j = 1, . . . , n 6. Stephane Beauregard, arXiv:quant-ph/0205095 (2003) L L ✓j = ✓0 leads to the most compelling results. [9] P. L. Montgomery. Modular Multiplication without Trial Division. Mathematics of Computation, 44(170):5 n 3(4), closed-form where ◆✓ a target ◆ rotation R'0 [✓T ], or The ! j goal is to✓implement APn (PDn) 2n n 12, analytic continuation Bibliography j ') L (~ = X {hk } 0 exp i j X k=1 k ( 1) 'hk X L T j equivalently Rk0 [✓T T ] with the replacement 'j ! 'j ( 1) . , fL ( ) = including the correct global phase, with a small k j k n+1 '0 , n > 12, conjectured error ToPn 2n arbitrary n, perturbative O(✏k=0). Specifically, we seek to eliminate the error of I: Summary of existing pulse sequences and our optimalon the faulty pulse M0 [✓T ] with a pulse sequence satisfying Table TABLE I: Comparison of known pulse sequences operating The primed sum enforces 1 h1 < · · · < hj L and T = ( + L)/2 . S = R0 [ ✏✓T ] + O(✏n+1 ) so that UT = S · M0 [✓T ] = R0 [✓T ] + O(✏ (1) n+1 ) imple- sequences PD,states and ToP. arbitrary AP, initial that suppress systematic amplitude errors to order n for arbitrary target rotation angles. Arbitrary accuracy generalizations are known or conjectured for all with the exception of SCROFULOUS. The sequences APn, PDn, The authors acknowledge the support of the NSF CUA, NSF iQuISE IGERT, IARPA QCS ORAQL, and NSF CCF-RQCC projects. www.postersession.com