One Complexity Theorist’s View of Quantum Computing Lance Fortnow NEC Research Institute

Comp.Theory FAQ 8. Complexity Theory (a) Lower Bounds (b) YACC (Yet Another Complexity Class) Our ability to understand and handle new models of computation comes from our experience studying previous notions. Case in Point: Quantum Computing

BQP: Yet Another Complexity Class Lance Fortnow NEC Research Institute

Quantum Computation A computation model based on quantum principles of physics. Ability to enter many parallel “states” and use interference to recover important information. Transformations must be unitary.

Dephysicfying Quantum To understand the computational powers of quantum computing, we should ignore the underlying physical model. Nondeterministic computation has no known underlying physical model yet we have a good understanding of its computational power.

The Quantum Class BQP The set of languages L such that there is a Polynomial-time Quantum Turing machine M such that for all strings x, If x is in L then the measured probability of acceptance of M on input x is at least 2/3. If x is not in L then the measured probability of acceptance of M on input x is at most 1/3.

Oddities of Quantum Computing Many Parallel States Similar to Probabilistic Computation. Interference Similar ideas in Counting Complexity. Unitary Transformations New and what makes quantum computing so hard to classify precisely.

A Product Machine Traditional nondeterministic Turing machine has a transition function Consider a generalized machine with transition function

The Computation Matrix The function  imposes a linear function mapping configurations to themselves. Consider the matrix M  capturing this linear function. The value of the computation after t steps is:

NP as Matrix Multiplication

#P as Matrix Multiplication

GapP as Matrix Multiplication

BPP as Matrix Multiplication

Small Changes

BQP as Matrix Multiplication

Questions Where’s the Physics? Where’s the ‘s? Where’s the real/complex numbers? Don’t we need reversibility? What if there is more than one accepting configuration? Where’s the measurements?

Where’s the Physics? Car makers have given us a model from which we can drive a car. Details of how the car works are not necessary.

Where’s ’s? Fancy way that physicists specify row and column vectors. Don’t need to deal with them when studying quantum complexity. Computer scientists like balance. What’s wrong with bra T and ket? Scares away newcomers.

Where’s the complex numbers? For BQP one can assume the transitions come from {-1,-4/5,-3/5,0, 3/5,4/5,1} instead of computable complex numbers. Noncomputable numbers allow encoding of noncomputable functions. Similar problem in classical model.

Don’t we need reversibility? The set of matrices M that preserve the L 2 norm are unitary: M(M * ) T is the identity. In particular M is invertible so the computation could be reversed. Reversibility is not a requirement of quantum computing but a consequence.

One accepting configuration? In most models, can assume one accepting configuration by having machine erase work tape and moving to single accept state. Not reversible process. Can be simulated in quantum with negligible additional error by writing answer and reversing the rest of the computation.

Where’s the measurements? Squaring value simulates process of measurement at end. Taking measurements during computation does not give additional power.

BQP - A good definition Simple and Robust. Based on a physical model. Contains interesting problems. Other Quantum Classes not as robust: EQP - Differences in set of allowable amplitudes may affect class. BQL - When measurements are made may affect class.

BQP as Matrix Multiplication

The Class AWPP

“Almost-Wide Probabilistic Polynomial Time” Previously Studied Fenner-Fortnow-Kurtz-Li Lide Li’s Thesis AWPP contains BQP

Properties of AWPP BQP  AWPP  PP  PSPACE AWPP is low for PP PP AWPP = PP For any L in AWPP, PP L = PP. There exists a relativized world where AWPP = P and the polynomial-time hierarchy is infinite.

Properties of BQP BQP  PP  PSPACE BQP is low for PP PP BQP = PP For any L in BQP, PP L = PP. There exists a relativized world where BQP = P and the polynomial-time hierarchy is infinite.

Diagram of Classes PSPACE P BQP AWPP PP PP-Low BPP NP PH

Diagram of Classes PSPACE P BQP AWPP PP PP-Low BPP NP PH

Diagram of Classes PSPACE P BQP AWPP PP PP-Low BPP NP PH

The Polynomial-Time Hierarchy Nondeterministic Computation is a misleading title. Really Existential. Similarly can have Universal Computation. Alternating TM - Switches back and forth between Existential and Universal. Unbounded Alternations - PSPACE Constant Alternations - PH

BQP in PH? Bernstein-Vazirani relativized language does not appear to sit in PH. It would if we allowed slightly more than polynomial-time or constant alternations. Suggestion: Try to show that BQP can be solved in quasipolynomial time and/or polylogarithmic alternations.

Diagram of Classes PSPACE P BQP AWPP PP PP-Low BPP NP PH

NP in BQP? Relative to a random oracle NP is in AWPP. Two problems: Random oracles do not give us a good view of the world. Need unitary transformations to get NP in BQP. Make it difficult to obtain bad consequences of NP in BQP.

Black Box Model

INPUT

N

Count only number of queries made. We do not care about computation time. Also known as decision tree or oracle model. Hard to define decision trees properly for quantum machines. NT

OR Function The OR function requires all N queries on some input of N bits for a deterministic machine. Adversary always answers zero on all queries. OR has small nondeterministic black box complexity (1 query).

Black Box Classes P – Polylogarithmic in N queries NP – Nondeterministic polylogarithmic in N queries The OR functions separates black box P from black box NP. How about BQP?

Black Box BQP The probability of acceptance of a black box BQP machine using t queries is a polynomial of degree at most 2t. Easy to see from Matrix Multiplication view of BQP.

BQP as Matrix Multiplication

The OR function The OR function has degree n. However a BQP black box need only approximate the OR function. Any polynomial that approximates the OR functions has degree  (  n).

Tightness of OR Any black box BQP machine must use  (  n) queries. OR function separates NP from BQP. Grover shows that O(  n) queries suffice to compute OR on a BQP machine.

General Result Any function f:{0,1} n  {0,1} that can be approximated by a degree d polynomial has a deterministic black box algorithm using O(d 6 ) queries. Due to Nisan-Szegedy, Beals-Buhrman- Cleve-Mosca-de Wolf.

BQP and P Every function computed by a BQP black box algorithm using t queries can be computed by a deterministic black box algorithm using O(t 6 ) queries. Black box BQP is the same as black box P.

Isn’t quantum better? What about Shor’s factoring, discrete logarithm, Deutch-Josza, Simon, etc. These have black box algorithms with limited input space. Deutch-Josza only separates all same from same number of zeros-ones. Factoring problem leads to black box with strong algebraic structure.

NP and BQP If BQP were to contain NP in the traditional model it would be because NP problems have a nice structure that BQP can take advantage of. To me this seems unlikely so I would conjecture that BQP cannot solve NP problems.

Is quantum computing useful? We can factor but … If the only uses of quantum computation remain discrete logarithms and factoring, it will likely become a special-purpose technique whose only raison d'etre is to thwart public key cryptosystems. (Peter Shor) Using tools of counting complexity, we have shown new bounds on power of quantum machines.

Conclusions Quantum Complexity very fascinating and worthy of future study. To study complexity of BQP forget the physics and their awful notation. Still seeking a definitive answer on usefulness of quantum computation. So far unable to use unitary property of BQP to help in classifying the class. Though useful in some oracle worlds.